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well_formed A → Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A _1:= P2 Γ A (A ---> A) A (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A))
(WFA : well_formed A): Γ
⊢i (A ---> (A ---> A) ---> A) --->
(A ---> A ---> A) ---> A ---> A
using BasicReasoning
Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A _1:= P2 Γ A (A ---> A) A (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A))
(WFA : well_formed A): Γ
⊢i (A ---> (A ---> A) ---> A) --->
(A ---> A ---> A) ---> A ---> A
using BasicReasoning _2:= P1 Γ A (A ---> A) (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A)): Γ ⊢i A ---> (A ---> A) ---> A
using BasicReasoning
Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A _1:= P2 Γ A (A ---> A) A (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A))
(WFA : well_formed A): Γ
⊢i (A ---> (A ---> A) ---> A) --->
(A ---> A ---> A) ---> A ---> A
using BasicReasoning _2:= P1 Γ A (A ---> A) (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A)): Γ ⊢i A ---> (A ---> A) ---> A
using BasicReasoning _3:= MP _2 _1: Γ ⊢i (A ---> A ---> A) ---> A ---> A
using BasicReasoning
Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A _1:= P2 Γ A (A ---> A) A (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A))
(WFA : well_formed A): Γ
⊢i (A ---> (A ---> A) ---> A) --->
(A ---> A ---> A) ---> A ---> A
using BasicReasoning _2:= P1 Γ A (A ---> A) (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A)): Γ ⊢i A ---> (A ---> A) ---> A
using BasicReasoning _3:= MP _2 _1: Γ ⊢i (A ---> A ---> A) ---> A ---> A
using BasicReasoning _4:= P1 Γ A A (WFA : well_formed A)
(WFA : well_formed A): Γ ⊢i A ---> A ---> A using BasicReasoning
Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A _1:= P2 Γ A (A ---> A) A (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A))
(WFA : well_formed A): Γ
⊢i (A ---> (A ---> A) ---> A) --->
(A ---> A ---> A) ---> A ---> A
using BasicReasoning _2:= P1 Γ A (A ---> A) (WFA : well_formed A)
(well_formed_imp WFA WFA
: well_formed (A ---> A)): Γ ⊢i A ---> (A ---> A) ---> A
using BasicReasoning _3:= MP _2 _1: Γ ⊢i (A ---> A ---> A) ---> A ---> A
using BasicReasoning _4:= P1 Γ A A (WFA : well_formed A)
(WFA : well_formed A): Γ ⊢i A ---> A ---> A using BasicReasoning _5:= MP _4 _3: Γ ⊢i A ---> A using BasicReasoning
Σ: Signature Γ: Theory l: hypotheses g: Pattern i, i': ProofInfo
ProofInfoLe i i'
→ Γ Ⱶ
l--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g
using i'
Σ: Signature Γ: Theory l: hypotheses g: Pattern i, i': ProofInfo
ProofInfoLe i i'
→ Γ Ⱶ
l--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g
using i'
Σ: Signature Γ: Theory l: hypotheses g: Pattern i, i': ProofInfo pile: ProofInfoLe i i' H: Γ Ⱶ
l--------------------------------------
g
using i
wf1: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i'
)) wf2: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i'
)))
mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i'
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i'
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i'
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i'
)
Σ: Signature Γ: Theory l: hypotheses g: Pattern i, i': ProofInfo pile: ProofInfoLe i i' H: mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i
) wf1: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i'
)) wf2: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i'
)))
mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i'
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i'
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i'
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i'
)
Σ: Signature Γ: Theory l: hypotheses g: Pattern i, i': ProofInfo pile: ProofInfoLe i i' H: Γ ⊢i foldr patt_imp g (patterns_of l) using i wf1: well_formed g wf2: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (patterns_of l)
using (ExGen := pi_generalized_evars,
SVSubst := pi_substituted_svars,
KT := pi_uses_kt, FP := pi_framing_patterns)
exact H.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H4:= P1 Γ
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) (A ---> B)
(well_formed_imp
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl))
: well_formed
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B)): Γ
⊢i (((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
(A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H4:= P1 Γ
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) (A ---> B)
(well_formed_imp
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl))
: well_formed
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B)): Γ
⊢i (((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
(A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H5:= MP H3 H4: Γ
⊢i (A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H4:= P1 Γ
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) (A ---> B)
(well_formed_imp
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl))
: well_formed
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B)): Γ
⊢i (((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
(A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H5:= MP H3 H4: Γ
⊢i (A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H6:= P2 Γ (A ---> B) ((A ---> B ---> ⊥) ---> A ---> B)
((A ---> B ---> ⊥) ---> A ---> ⊥)
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB)
: well_formed
((A ---> B ---> ⊥) ---> A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl)
: well_formed
((A ---> B ---> ⊥) ---> A ---> ⊥)): Γ
⊢i ((A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H4:= P1 Γ
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) (A ---> B)
(well_formed_imp
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl))
: well_formed
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B)): Γ
⊢i (((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
(A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H5:= MP H3 H4: Γ
⊢i (A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H6:= P2 Γ (A ---> B) ((A ---> B ---> ⊥) ---> A ---> B)
((A ---> B ---> ⊥) ---> A ---> ⊥)
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB)
: well_formed
((A ---> B ---> ⊥) ---> A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl)
: well_formed
((A ---> B ---> ⊥) ---> A ---> ⊥)): Γ
⊢i ((A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H7:= MP H5 H6: Γ
⊢i ((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H4:= P1 Γ
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) (A ---> B)
(well_formed_imp
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl))
: well_formed
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B)): Γ
⊢i (((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
(A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H5:= MP H3 H4: Γ
⊢i (A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H6:= P2 Γ (A ---> B) ((A ---> B ---> ⊥) ---> A ---> B)
((A ---> B ---> ⊥) ---> A ---> ⊥)
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB)
: well_formed
((A ---> B ---> ⊥) ---> A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl)
: well_formed
((A ---> B ---> ⊥) ---> A ---> ⊥)): Γ
⊢i ((A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H7:= MP H5 H6: Γ
⊢i ((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H8:= P1 Γ (A ---> B) (A ---> B ---> ⊥)
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥)): Γ
⊢i (A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H1:= P2 Γ A B ⊥ (WFA : well_formed A)
(WFB : well_formed B) (erefl : well_formed ⊥): Γ
⊢i (A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥ using BasicReasoning H2:= P2 Γ (A ---> B ---> ⊥) (A ---> B) (A ---> ⊥)
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA erefl
: well_formed (A ---> ⊥)): Γ
⊢i ((A ---> B ---> ⊥) --->
(A ---> B) ---> A ---> ⊥) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H3:= MP H1 H2: Γ
⊢i ((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H4:= P1 Γ
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) (A ---> B)
(well_formed_imp
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl))
: well_formed
(((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥))
(well_formed_imp WFA WFB
: well_formed (A ---> B)): Γ
⊢i (((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
(A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H5:= MP H3 H4: Γ
⊢i (A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H6:= P2 Γ (A ---> B) ((A ---> B ---> ⊥) ---> A ---> B)
((A ---> B ---> ⊥) ---> A ---> ⊥)
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA WFB)
: well_formed
((A ---> B ---> ⊥) ---> A ---> B))
(well_formed_imp
(well_formed_imp WFA
(well_formed_imp WFB erefl))
(well_formed_imp WFA erefl)
: well_formed
((A ---> B ---> ⊥) ---> A ---> ⊥)): Γ
⊢i ((A ---> B) --->
((A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥) --->
((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H7:= MP H5 H6: Γ
⊢i ((A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B) --->
(A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning H8:= P1 Γ (A ---> B) (A ---> B ---> ⊥)
(well_formed_imp WFA WFB
: well_formed (A ---> B))
(well_formed_imp WFA (well_formed_imp WFB erefl)
: well_formed (A ---> B ---> ⊥)): Γ
⊢i (A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> B
using BasicReasoning H9:= MP H8 H7: Γ
⊢i (A ---> B) --->
(A ---> B ---> ⊥) ---> A ---> ⊥
using BasicReasoning
Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
exact H9.Defined.
Σ: Signature Γ: Theory A: Pattern
well_formed A
→ Γ ⊢i (A ---> ! A) ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A
→ Γ ⊢i (A ---> ! A) ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i (A ---> ! A) ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁ ---> (A ---> ! A) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁using BasicReasoning
apply (@A_impl_A _ _ A WFA).
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i (A ---> A) ---> (A ---> ! A) ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i (A ---> A) ---> (A ---> ! A) ---> ! A
using BasicReasoning
apply (@P4m _ _ A A WFA WFA).}Defined.
Σ: Signature Γ: Theory A, B, C: Pattern
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning
Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁ ---> (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0using BasicReasoning
apply(P1 _ B A ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i (B ---> A ---> B) ---> ?ϕ₁using BasicReasoning
apply(P1 _ (B ---> A ---> B) (A ---> B ---> C) ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ((A ---> B ---> C) ---> B ---> A ---> B) --->
(A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ?ϕ₁ --->
((A ---> B ---> C) ---> B ---> A ---> B) --->
(A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1 ---> ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁2using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁2 ---> ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁2using BasicReasoning
apply (@A_impl_A _ _ ABC ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i (ABC ---> ABC) ---> ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁2using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁2using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁3using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁3 ---> ?ϕ₁2using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁3using BasicReasoning
apply(P2 _ A B C ltac:(wf_auto2) ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ((A ---> B ---> C) ---> (A ---> B) ---> A ---> C) --->
?ϕ₁2using BasicReasoning
unshelve (eapply(P1 _ _ (A ---> B ---> C) _ _)); wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ((A ---> B ---> C) --->
(A ---> B ---> C) ---> (A ---> B) ---> A ---> C) --->
(ABC ---> ABC) ---> ?ϕ₁1using BasicReasoning
apply P2; wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i (ABC ---> (A ---> B) ---> A ---> C) ---> ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ?ϕ₁1 --->
(ABC ---> (A ---> B) ---> A ---> C) ---> ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1using BasicReasoning
apply t1.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ((A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C) --->
(ABC ---> (A ---> B) ---> A ---> C) ---> ?ϕ₁0using BasicReasoning
apply(P2 _ ABC ((A ---> B) ---> (A ---> C)) (B ---> (A ---> B) ---> (A ---> C)) ltac:(wf_auto2) ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i (ABC ---> B ---> (A ---> B) ---> A ---> C) ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ?ϕ₁0 --->
(ABC ---> B ---> (A ---> B) ---> A ---> C) ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1 ---> ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ ⊢i ?ϕ₁1using BasicReasoning
apply(P2 _ B (A ---> B) (A ---> C) ltac:(wf_auto2) ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i ((B ---> (A ---> B) ---> A ---> C) --->
(B ---> A ---> B) ---> B ---> A ---> C) ---> ?ϕ₁0using BasicReasoning
apply(P1 _ _ ABC); wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i (ABC --->
(B ---> (A ---> B) ---> A ---> C) --->
(B ---> A ---> B) ---> B ---> A ---> C) --->
(ABC ---> B ---> (A ---> B) ---> A ---> C) ---> ?ϕ₁using BasicReasoning
apply P2; wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C t1:= MP
(P1 Γ ((A ---> B) ---> A ---> C) B
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)
: well_formed ((A ---> B) ---> A ---> C))
(WFB : well_formed B))
(P1 Γ
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C)
(A ---> B ---> C)
(well_formed_imp
(well_formed_imp (well_formed_imp WFA WFB)
(well_formed_imp WFA WFC))
(well_formed_imp WFB
(well_formed_imp
(well_formed_imp WFA WFB)
(well_formed_imp WFA WFC)))
: well_formed
(((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C))
(well_formed_imp WFA
(well_formed_imp WFB WFC)
: well_formed (A ---> B ---> C))): Γ
⊢i (A ---> B ---> C) --->
((A ---> B) ---> A ---> C) --->
B ---> (A ---> B) ---> A ---> C
using BasicReasoning ABC:= A ---> B ---> C: Pattern
Γ
⊢i (ABC ---> (B ---> A ---> B) ---> B ---> A ---> C) --->
((A ---> B ---> C) ---> B ---> A ---> B) --->
(A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
apply P2; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A ---> B ---> C using i
→ Γ ⊢i B ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A ---> B ---> C using i
→ Γ ⊢i B ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B ---> C using i
Γ ⊢i B ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B ---> C using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B ---> C using i
Γ ⊢i ?ϕ₁ ---> B ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B ---> C using i
Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B ---> C using i
Γ ⊢i (A ---> B ---> C) ---> B ---> A ---> C
using BasicReasoning
apply reorder; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, C: Pattern
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i (A ---> B) ---> (B ---> C) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i (A ---> B) ---> (B ---> C) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i (A ---> B) ---> (B ---> C) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i (B ---> C) ---> (A ---> B) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁ ---> (B ---> C) ---> (A ---> B) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁using BasicReasoning
apply(P1 _ (B ---> C) A); wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ
⊢i ((B ---> C) ---> A ---> B ---> C) --->
(B ---> C) ---> (A ---> B) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ
⊢i ?ϕ₁ --->
((B ---> C) ---> A ---> B ---> C) --->
(B ---> C) ---> (A ---> B) ---> A ---> C
using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ ⊢i ?ϕ₁0using BasicReasoning
apply (P2 _ A B C); wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ
⊢i ((A ---> B ---> C) ---> (A ---> B) ---> A ---> C) --->
?ϕ₁using BasicReasoning
apply (P1 _ ((A ---> B ---> C) ---> (A ---> B) ---> A ---> C) (B ---> C)); wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern WFA: well_formed A WFB: well_formed B WFC: well_formed C
Γ
⊢i ((B ---> C) --->
(A ---> B ---> C) ---> (A ---> B) ---> A ---> C) --->
((B ---> C) ---> A ---> B ---> C) --->
(B ---> C) ---> (A ---> B) ---> A ---> C
using BasicReasoning
apply P2; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A ---> B using i
→ Γ ⊢i B ---> C using i → Γ ⊢i A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A ---> B using i
→ Γ ⊢i B ---> C using i → Γ ⊢i A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i ?ϕ₁ ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i ?ϕ₁using i
exact H2.
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i (A ---> B) ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i ?ϕ₁ ---> (A ---> B) ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i ?ϕ₁using i
exact H3.
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i (B ---> C) ---> (A ---> B) ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i (A ---> B) ---> (B ---> C) ---> A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo H: well_formed A H0: well_formed B H1: well_formed C H2: Γ ⊢i A ---> B using i H3: Γ ⊢i B ---> C using i
Γ ⊢i (A ---> B) ---> (B ---> C) ---> A ---> C
using BasicReasoning
apply syllogism; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> (A ---> B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> (A ---> B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> (A ---> B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁ ---> A ---> (A ---> B) ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁using BasicReasoning
apply (P1 _ A (A ---> B) ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ
⊢i (A ---> (A ---> B) ---> A) --->
A ---> (A ---> B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ
⊢i ?ϕ₁ --->
(A ---> (A ---> B) ---> A) --->
A ---> (A ---> B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?ϕ₁0using BasicReasoning
apply (@A_impl_A _ _ (A ---> B) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ((A ---> B) ---> A ---> B) ---> ?ϕ₁using BasicReasoning
eapply (P2 _ (A ---> B) A B ltac:(wf_auto2) ltac:(wf_auto2) ltac:(wf_auto2)).
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ
⊢i (((A ---> B) ---> A) ---> (A ---> B) ---> B) --->
(A ---> (A ---> B) ---> A) --->
A ---> (A ---> B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ
⊢i (A ---> (A ---> B) ---> A) --->
(((A ---> B) ---> A) ---> (A ---> B) ---> B) --->
A ---> (A ---> B) ---> B using BasicReasoning
apply syllogism; wf_auto2.Defined.
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A ---> ! ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A ---> ! ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i A ---> ! ! A using BasicReasoning
apply modus_ponens; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (! A ---> ! B) ---> B ---> A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (! A ---> ! B) ---> B ---> A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m3: Γ ⊢i (B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m3: Γ ⊢i (B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m4: Γ
⊢i ((B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m3: Γ ⊢i (B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m4: Γ
⊢i ((B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m5: Γ
⊢i ((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m3: Γ ⊢i (B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m4: Γ
⊢i ((B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m5: Γ
⊢i ((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m6: Γ
⊢i (((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
(((A ---> ⊥) ---> B ---> ⊥) --->
B ---> (A ---> ⊥) ---> ⊥) --->
((A ---> ⊥) ---> B ---> ⊥) ---> B ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m3: Γ ⊢i (B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m4: Γ
⊢i ((B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m5: Γ
⊢i ((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m6: Γ
⊢i (((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
(((A ---> ⊥) ---> B ---> ⊥) --->
B ---> (A ---> ⊥) ---> ⊥) --->
((A ---> ⊥) ---> B ---> ⊥) ---> B ---> A
using BasicReasoning m7: Γ
⊢i (((A ---> ⊥) ---> B ---> ⊥) --->
B ---> (A ---> ⊥) ---> ⊥) --->
((A ---> ⊥) ---> B ---> ⊥) ---> B ---> A
using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m0: Γ
⊢i (((A ---> ⊥) ---> ⊥) ---> A) --->
B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m1: Γ
⊢i (B ---> ((A ---> ⊥) ---> ⊥) ---> A) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m2: Γ ⊢i B ---> ((A ---> ⊥) ---> ⊥) ---> A
using BasicReasoning m3: Γ ⊢i (B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m4: Γ
⊢i ((B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m5: Γ
⊢i ((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A
using BasicReasoning m6: Γ
⊢i (((A ---> ⊥) ---> B ---> ⊥) --->
(B ---> (A ---> ⊥) ---> ⊥) ---> B ---> A) --->
(((A ---> ⊥) ---> B ---> ⊥) --->
B ---> (A ---> ⊥) ---> ⊥) --->
((A ---> ⊥) ---> B ---> ⊥) ---> B ---> A
using BasicReasoning m7: Γ
⊢i (((A ---> ⊥) ---> B ---> ⊥) --->
B ---> (A ---> ⊥) ---> ⊥) --->
((A ---> ⊥) ---> B ---> ⊥) ---> B ---> A
using BasicReasoning m8: Γ
⊢i ((A ---> ⊥) ---> B ---> ⊥) --->
B ---> (A ---> ⊥) ---> ⊥ using BasicReasoning
Γ ⊢i (! A ---> ! B) ---> B ---> A using BasicReasoning
apply (MP m8 m7).Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: ?Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: ?Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: ?Γ0 ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: ?Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: ?Γ0 ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ1 ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: ?Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ0 ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': ?Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': ?Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: ?Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': ?Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: ?Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': ?Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': ?Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: ?Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning t6: Γ ⊢i A ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning t6: Γ ⊢i A ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t7: Γ ⊢i A ---> B ---> ! (! ! A ---> ! B)
using BasicReasoning
Γ ⊢i A ---> B ---> A and B using BasicReasoning
apply t7.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning
well_formed
(((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning
well_formed (! ! A ---> ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning
well_formed (B ---> (! ! A ---> ! B) ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed (! ! A ---> ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed (A ---> (! ! A ---> ! B) ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed ((! ! A ---> ! B) ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed ((! ! A ---> ! B) ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning
well_formed
(((! ! A ---> ! B) ---> A) ---> ! (! ! A ---> ! B))
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning
well_formed (B ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning
well_formed ((! ! A ---> ! B) ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed
((B ---> A) ---> B ---> (! ! A ---> ! B) ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed ((! ! A ---> ! B) ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed (! (! ! A ---> ! B))
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning
well_formed
((B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B))
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning
well_formed (B ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning
well_formed (B ---> (! ! A ---> ! B) ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning t6: Γ ⊢i A ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning t6: Γ ⊢i A ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed (B ---> (! ! A ---> ! B) ---> A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B tB: Γ ⊢i B ---> B using BasicReasoning t1: Γ
⊢i B --->
((! ! A ---> ! B) ---> A ---> ⊥) --->
((! ! A ---> ! B) ---> A) --->
(! ! A ---> ! B) ---> ⊥ using BasicReasoning t2: Γ ⊢i B ---> B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t3'': Γ ⊢i B ---> A ---> (! ! A ---> ! B) ---> A
using BasicReasoning t4: Γ ⊢i B ---> (! ! A ---> ! B) ---> ! A
using BasicReasoning t5'': Γ
⊢i B --->
((! ! A ---> ! B) ---> A) --->
! (! ! A ---> ! B) using BasicReasoning tA: Γ ⊢i A ---> B ---> A using BasicReasoning tB': Γ ⊢i A ---> B ---> B using BasicReasoning t3': Γ
⊢i (B ---> A) --->
B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t3: Γ
⊢i A --->
(B ---> A) ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning t5': Γ
⊢i (B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B)
using BasicReasoning t5: Γ
⊢i A --->
(B ---> (! ! A ---> ! B) ---> A) --->
B ---> ! (! ! A ---> ! B) using BasicReasoning t6: Γ ⊢i A ---> B ---> (! ! A ---> ! B) ---> A
using BasicReasoning
well_formed (B ---> ! (! ! A ---> ! B))
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A using i
→ Γ ⊢i B using i → Γ ⊢i A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A using i
→ Γ ⊢i B using i → Γ ⊢i A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i ?ϕ₁ ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i ?ϕ₁using i
exact H0.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i ?ϕ₁ ---> B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i ?ϕ₁using i
exact H.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i A ---> B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i H0: Γ ⊢i B using i
Γ ⊢i A ---> B ---> A and B using BasicReasoning
apply conj_intro; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ well_formed D
→ Γ ⊢i A ---> B ---> C using i
→ Γ ⊢i C ---> D using i
→ Γ ⊢i A ---> B ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ well_formed D
→ Γ ⊢i A ---> B ---> C using i
→ Γ ⊢i C ---> D using i
→ Γ ⊢i A ---> B ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i A ---> B ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁ ---> A ---> B ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁using i
exact H.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i (A ---> B ---> C) ---> A ---> B ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁ ---> (A ---> B ---> C) ---> A ---> B ---> D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁0using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁0using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁1using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁1 ---> ?ϕ₁0using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁1using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁2using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁2 ---> ?ϕ₁1using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ?ϕ₁2using i
exact H0.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i (C ---> D) ---> ?ϕ₁1using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i (C ---> D) ---> ?ϕ₁1using BasicReasoning
eapply (P1 _ (C ---> D) B _ _).
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i (B ---> C ---> D) ---> ?ϕ₁0using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i (B ---> C ---> D) ---> ?ϕ₁0using BasicReasoning
eapply (P2 _ B C D _ _ _).
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ((B ---> C) ---> B ---> D) ---> ?ϕ₁using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ ⊢i ((B ---> C) ---> B ---> D) ---> ?ϕ₁using BasicReasoning
eapply (P1 _ ((B ---> C) ---> B ---> D) A _ _).
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ
⊢i (A ---> (B ---> C) ---> B ---> D) --->
(A ---> B ---> C) ---> A ---> B ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
Γ
⊢i (A ---> (B ---> C) ---> B ---> D) --->
(A ---> B ---> C) ---> A ---> B ---> D
using BasicReasoning
eapply (P2 _ A (B ---> C) (B ---> D) _ _ _).
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed (C ---> D)
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed B
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed B
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed C
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed D
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed ((B ---> C) ---> B ---> D)
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed A
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed A
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed (B ---> C)
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B ---> C using i H0: Γ ⊢i C ---> D using i
well_formed (B ---> D)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i ⊥ ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i ⊥ ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ⊥ ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁ ---> ⊥ ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁1 ---> ?ϕ₁0using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁2using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁2 ---> ?ϕ₁1using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ?ϕ₁2using BasicReasoning
eapply (P1 _ Bot Bot _ _).
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i (⊥ ---> ⊥) ---> ?ϕ₁using BasicReasoning
eapply (P1 _ (Bot ---> Bot) (A ---> Bot) _ _).
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ((A ---> ⊥) ---> ⊥ ---> ⊥) ---> ⊥ ---> A
using BasicReasoning
eapply (@P4 _ _ A Bot _ _).
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed (⊥ ---> ⊥)
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed (A ---> ⊥)
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed A
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed ⊥
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> (! B ---> ! A) ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> (! B ---> ! A) ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> (! B ---> ! A) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! B ---> ! A) ---> A ---> B using BasicReasoning
apply P4; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i B ---> A or B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i B ---> A or B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i B ---> A or B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i B ---> A or B using BasicReasoning
apply P1; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> A or B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> A or B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> A or B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed ?C
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> ! A ---> ?Cusing BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?C ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> ! A ---> ?Cusing BasicReasoning
apply modus_ponens; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed ⊥
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ⊥ ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ⊥ ---> B using BasicReasoning
apply bot_elim; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed ⊥
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B → Γ ⊢i B using i → Γ ⊢i A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B → Γ ⊢i B using i → Γ ⊢i A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i ?ϕ₁ ---> A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i ?ϕ₁using i
exact HB.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i B ---> A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i B ---> A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HB: Γ ⊢i B using i
Γ ⊢i B ---> A or B using BasicReasoning
apply disj_right_intro; wf_auto2.}Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B → Γ ⊢i A using i → Γ ⊢i A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B → Γ ⊢i A using i → Γ ⊢i A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HA: Γ ⊢i A using i
Γ ⊢i A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HA: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HA: Γ ⊢i A using i
Γ ⊢i ?ϕ₁ ---> A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HA: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
exact HA.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HA: Γ ⊢i A using i
Γ ⊢i A ---> A or B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo HwfA: well_formed A HwfB: well_formed B HA: Γ ⊢i A using i
Γ ⊢i A ---> A or B using BasicReasoning
apply disj_left_intro; assumption.Defined.
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i ! ! A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i ! ! A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Γ ⊢i ! ! A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
well_formed A
exact WFA.Defined.
Σ: Signature Γ: Theory A: Pattern i: ProofInfo
well_formed A → Γ ⊢i ! ! A using i → Γ ⊢i A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo
well_formed A → Γ ⊢i ! ! A using i → Γ ⊢i A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i ?ϕ₁ ---> A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i ?ϕ₁using i
apply nnA.
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i ! ! A ---> A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i ! ! A ---> A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
Γ ⊢i ! ! A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern i: ProofInfo wfA: well_formed A nnA: Γ ⊢i ! ! A using i
well_formed A
exact wfA.}Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (! ! A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (! ! A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! ! A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! ! A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed ?B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! ! A ---> ! ! B) ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?B ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! ! A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! ! A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i ! ! A ---> ! ! B using i
→ Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i ! ! A ---> ! ! B using i
→ Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! ! A ---> ! ! B using i
Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! ! A ---> ! ! B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! ! A ---> ! ! B using i
Γ ⊢i ?ϕ₁ ---> A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! ! A ---> ! ! B using i
Γ ⊢i ?ϕ₁using i
exact H.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! ! A ---> ! ! B using i
Γ ⊢i (! ! A ---> ! ! B) ---> A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! ! A ---> ! ! B using i
Γ ⊢i (! ! A ---> ! ! B) ---> A ---> B
using BasicReasoning
apply double_neg_elim; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (! ! A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (! ! A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (A ---> B)
shelve.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (! ! A ---> B)
shelve.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (! ! A ---> ! ! B)
shelve.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning
assumption.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
assumption.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning
well_formed (B ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning
well_formed (! ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning
well_formed (! ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning
well_formed (! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (! ! A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B S1:= syllogism Γ ?Goal?Goal0?Goal1: Γ
⊢i (! ! A ---> A) --->
(A ---> B) ---> ! ! A ---> B
using BasicReasoning MP1:= MP (not_not_elim Γ ?Goal2) S1: Γ ⊢i (A ---> B) ---> ! ! A ---> B using BasicReasoning NNB:= not_not_intro Γ ?Goal3: Γ ⊢i B ---> ! ! B using BasicReasoning P1:= ProofMode_propositional.P1 Γ (B ---> ! ! B)
(! ! A) ?Goal4?Goal5: Γ ⊢i (B ---> ! ! B) ---> ! ! A ---> B ---> ! ! B
using BasicReasoning MP2:= MP NNB P1: Γ ⊢i ! ! A ---> B ---> ! ! B using BasicReasoning P2':= P2 Γ (! ! A) B (! ! B) ?Goal6?Goal7?Goal8: Γ ⊢i (! ! A ---> B ---> ! ! B) ---> (! ! A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning MP3:= MP MP2 P2': Γ ⊢i (! ! A ---> B) ---> ! ! A ---> ! ! B using BasicReasoning
well_formed (! ! A ---> ! ! B)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> B) ---> ! B ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> B) ---> ! B ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (A ---> B) ---> ! B ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Γ ⊢i (A ---> B) ---> ! B ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (! ! A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (! B ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (A ---> B)
shelve.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (! ! A ---> ! ! B)
shelve.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (! B ---> ! A)
shelve.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Γ ⊢i (A ---> B) ---> ! ! A ---> ! ! B
using BasicReasoning
apply @not_not_impl_intro; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
exact m. (* apply (P4 _ _ _). shelve. shelve. *)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (! ! A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B m: Γ ⊢i (! ! A ---> ! ! B) ---> ! B ---> ! A
using BasicReasoning
well_formed (! B ---> ! A)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A or B using i → Γ ⊢i B or A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A or B using i → Γ ⊢i B or A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A or B using i
Γ ⊢i B or A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i
Γ ⊢i ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A
using BasicReasoning
Γ ⊢i ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A
using BasicReasoning NNI: Γ ⊢i B ---> ! ! B using BasicReasoning
Γ ⊢i ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using BasicReasoning
Γ ⊢i ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i
Γ ⊢i ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i SI: Γ ⊢i ! A ---> ! ! B using i
Γ ⊢i ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i SI: Γ ⊢i ! A ---> ! ! B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i SI: Γ ⊢i ! A ---> ! ! B using i
Γ ⊢i ?ϕ₁ ---> ! B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i SI: Γ ⊢i ! A ---> ! ! B using i
Γ ⊢i ?ϕ₁using i
exact SI.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i SI: Γ ⊢i ! A ---> ! ! B using i
Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i
exact P4.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A
using BasicReasoning
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> B using i P4: Γ ⊢i (! A ---> ! ! B) ---> ! B ---> A using i NNI: Γ ⊢i B ---> ! ! B using i
well_formed (! ! B)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A: Pattern i: ProofInfo
well_formed A → Γ ⊢i A using i → Γ ⊢i ! ! A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo
well_formed A → Γ ⊢i A using i → Γ ⊢i ! ! A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ! ! A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁ ---> (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i A ---> (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i A ---> (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i A ---> (A ---> ⊥) ---> ⊥ using BasicReasoning
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
well_formed A
exact WFA.}Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i ! A ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i ! A ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! A ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed ?B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! A ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ?B ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! A ---> ! B ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B ---> ! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
all: wf_auto2.Defined.
Σ: Signature Γ: Theory phi: Pattern
well_formed phi → Γ ⊢i ⊥ ---> phi using BasicReasoning
Σ: Signature Γ: Theory phi: Pattern
well_formed phi → Γ ⊢i ⊥ ---> phi using BasicReasoning
apply bot_elim.Defined.
Σ: Signature Γ: Theory A: Pattern i: ProofInfo
well_formed A → Γ ⊢i A using i → Γ ⊢i ! ! A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo
well_formed A → Γ ⊢i A using i → Γ ⊢i ! ! A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ! ! A using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁ ---> (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i A ---> (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i A ---> (A ---> ⊥) ---> ⊥ using i
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
Γ ⊢i A ---> (A ---> ⊥) ---> ⊥ using BasicReasoning
Σ: Signature Γ: Theory A: Pattern i: ProofInfo WFA: well_formed A H: Γ ⊢i A using i
well_formed A
exact WFA.}Defined.
Σ: Signature G: Theory A: Pattern i: ProofInfo
well_formed A
→ G ⊢i A using i
→ G ⊢i A ---> ⊥ using i → G ⊢i ⊥ using i
Σ: Signature G: Theory A: Pattern i: ProofInfo
well_formed A
→ G ⊢i A using i
→ G ⊢i A ---> ⊥ using i → G ⊢i ⊥ using i
Σ: Signature G: Theory A: Pattern i: ProofInfo WFA: well_formed A H: G ⊢i A using i H0: G ⊢i A ---> ⊥ using i
G ⊢i ⊥ using i
Σ: Signature G: Theory A: Pattern i: ProofInfo WFA: well_formed A H: G ⊢i A using i H0: G ⊢i A ---> ⊥ using i
G ⊢i ?ϕ₁using i
Σ: Signature G: Theory A: Pattern i: ProofInfo WFA: well_formed A H: G ⊢i A using i H0: G ⊢i A ---> ⊥ using i
G ⊢i ?ϕ₁ ---> ⊥ using i
Σ: Signature G: Theory A: Pattern i: ProofInfo WFA: well_formed A H: G ⊢i A using i H0: G ⊢i A ---> ⊥ using i
G ⊢i A ---> ⊥ using i
apply H0.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
Γ ⊢i A ---> B using i → Γ ⊢i ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
Γ ⊢i A ---> B using i → Γ ⊢i ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i
Γ ⊢i ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B)
Γ ⊢i ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A
Γ ⊢i ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A wfB: well_formed B
Γ ⊢i ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A wfB: well_formed B
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A wfB: well_formed B
Γ ⊢i ?ϕ₁ ---> ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A wfB: well_formed B
Γ ⊢i ?ϕ₁ ---> ! B ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A wfB: well_formed B
Γ ⊢i ?ϕ₁ ---> ! B ---> ! A using BasicReasoning
apply contraposition; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo H: Γ ⊢i A ---> B using i wf: well_formed (A ---> B) wfA: well_formed A wfB: well_formed B
Γ ⊢i A ---> B using i
apply H.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i (A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (A ---> ! ! B) ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! ! B ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! ! B ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> (! ! B ---> B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> (! ! B ---> B) ---> A ---> B
using BasicReasoning
apply syllogism; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> A ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (A ---> ! ! B) ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
well_formed (A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
well_formed ?ϕ₁
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i (A ---> ! ! B) ---> ?ϕ₁ ---> A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
well_formed (A ---> ! ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
well_formed (! ! B ---> B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
well_formed (A ---> B)
all: wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H0: Γ ⊢i ! ! B ---> B using BasicReasoning H1: Γ
⊢i (A ---> ! ! B) --->
(! ! B ---> B) ---> A ---> B
using BasicReasoning
Γ ⊢i ! ! B ---> B using BasicReasoning
apply H0.Defined.
Σ: Signature Γ: Theory A, B, B': Pattern
well_formed A
→ well_formed B
→ well_formed B'
→ Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using BasicReasoning
Σ: Signature Γ: Theory A, B, B': Pattern
well_formed A
→ well_formed B
→ well_formed B'
→ Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using BasicReasoning
Σ: Signature Γ: Theory A, B, B': Pattern wfA: well_formed A wfB: well_formed B wfB': well_formed B'
Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using BasicReasoning
Σ: Signature Γ: Theory A, B, B': Pattern wfA: well_formed A wfB: well_formed B wfB': well_formed B'
Γ ⊢i (A ---> B) ---> (B ---> B') ---> A ---> B'
using BasicReasoning
apply syllogism; wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed B'
→ Γ ⊢i B ---> B' using i
→ Γ ⊢i (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed B'
→ Γ ⊢i B ---> B' using i
→ Γ ⊢i (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i
Γ ⊢i (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i
Γ ⊢i (A ---> B) ---> (B ---> B') ---> A ---> B'
using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i H1: Γ ⊢i (A ---> B) ---> (B ---> B') ---> A ---> B'
using i
Γ ⊢i (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i
Γ ⊢i (A ---> B) ---> (B ---> B') ---> A ---> B'
using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i
Γ ⊢i (A ---> B) ---> (B ---> B') ---> A ---> B'
using BasicReasoning
apply syllogism; wf_auto2.
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i H1: Γ ⊢i (A ---> B) ---> (B ---> B') ---> A ---> B'
using i
Γ ⊢i (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i H1: Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using i
Γ ⊢i (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i H1: Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i H1: Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using i
Γ ⊢i ?ϕ₁ ---> (A ---> B) ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfB': well_formed B' BimpB': Γ ⊢i B ---> B' using i H1: Γ ⊢i (B ---> B') ---> (A ---> B) ---> A ---> B'
using i
Γ ⊢i B ---> B' using i
apply BimpB'.Defined.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g'
Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g'
Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory g, g': Pattern wfl: Pattern.wf [] wfg: well_formed g wfg': well_formed g'
Γ
⊢i (g ---> g') --->
foldr patt_imp g [] ---> foldr patt_imp g' []
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory g, g': Pattern wfl: Pattern.wf [] wfg: well_formed g wfg': well_formed g'
Γ
⊢i (g ---> g') --->
foldr patt_imp g [] ---> foldr patt_imp g' []
using BasicReasoning
Σ: Signature Γ: Theory g, g': Pattern wfl: Pattern.wf [] wfg: well_formed g wfg': well_formed g'
well_formed
(foldr patt_imp g [] ---> foldr patt_imp g' [])
wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l)
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfl: well_formed a = true
∧ (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l)
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfl: well_formed a = true
∧ (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end) (map well_formed l) = true wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l)
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end) (map well_formed l) = true wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l)
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
(* I do not know how to fold it, so I just assert & clear. *)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end) (map well_formed l) = true wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Pattern.wf l
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ
⊢i (g ---> g') --->
foldr patt_imp g (a :: l) --->
foldr patt_imp g' (a :: l) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ
⊢i (g ---> g') --->
(a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g' l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (g ---> g')
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed ?B
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed
((a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g' l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ ⊢i (g ---> g') ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ
⊢i ?B --->
(a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g' l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (g ---> g')
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed
(foldr patt_imp g l ---> foldr patt_imp g' l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed
((a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g' l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed a
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (foldr patt_imp g l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (foldr patt_imp g' l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (g ---> g')
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed
(foldr patt_imp g l ---> foldr patt_imp g' l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed
((a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g' l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed a
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (foldr patt_imp g l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, g': Pattern wfa: well_formed a = true wfg: well_formed g wfg': well_formed g' IHl: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning wfl': Pattern.wf (a :: l) wfl'': Pattern.wf l
well_formed (foldr patt_imp g' l)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ ⊢i g ---> g' using i
→ Γ
⊢i foldr patt_imp g l ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ ⊢i g ---> g' using i
→ Γ
⊢i foldr patt_imp g l ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ ⊢i foldr patt_imp g l ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g l ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g l ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
apply prf_weaken_conclusion_iter; wf_auto2.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ ⊢i g ---> g' using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i
Γ ⊢i g ---> g' using i
apply gimpg'.}Defined.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ ⊢i g ---> g' using i
→ Γ ⊢i foldr patt_imp g l using i
→ Γ ⊢i foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ ⊢i g ---> g' using i
→ Γ ⊢i foldr patt_imp g l using i
→ Γ ⊢i foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i ?ϕ₁using i
apply gimpg'.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i (g ---> g') ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i ?ϕ₁ ---> (g ---> g') ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ
⊢i foldr patt_imp g l --->
(g ---> g') ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
Pattern.wf l
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
well_formed g
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' gimpg': Γ ⊢i g ---> g' using i H: Γ ⊢i foldr patt_imp g l using i
well_formed g'
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed B'
→ Γ ⊢i B ---> B' using i
→ Γ ⊢i A ---> B using i → Γ ⊢i A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed B'
→ Γ ⊢i B ---> B' using i
→ Γ ⊢i A ---> B using i → Γ ⊢i A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
Γ ⊢i A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
Γ ⊢i ?ϕ₁ ---> A ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
Γ ⊢i A ---> ?Busing i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed ?B
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed B'
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
Γ ⊢i ?B ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed B
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed B'
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
Γ ⊢i B ---> B' using i
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed B
Σ: Signature Γ: Theory A, B, B': Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFB': well_formed B' H: Γ ⊢i B ---> B' using i H0: Γ ⊢i A ---> B using i
well_formed B'
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, A', B: Pattern
well_formed A
→ well_formed A'
→ well_formed B
→ Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, A', B: Pattern
well_formed A
→ well_formed A'
→ well_formed B
→ Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, A', B: Pattern wfA: well_formed A wfA': well_formed A' wfB: well_formed B
Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using BasicReasoning
apply syllogism; wf_auto2.Defined.
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf l₁ wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ([] ++ h :: l₂) --->
foldr patt_imp g ([] ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf (a :: l₁) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ([] ++ h :: l₂) --->
foldr patt_imp g ([] ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
Γ
⊢i (h' ---> h) --->
(h ---> foldr patt_imp g l₂) --->
h' ---> foldr patt_imp g l₂ using BasicReasoning
Σ: Signature Γ: Theory l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
well_formed h
Σ: Signature Γ: Theory l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
well_formed h'
Σ: Signature Γ: Theory l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g
well_formed (foldr patt_imp g l₂)
all: wf_auto2.
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf (a :: l₁) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf (a :: l₁) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf (a :: l₁) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: Pattern.wf (a :: l₁) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: foldr andb true (map well_formed (a :: l₁)) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: well_formed a &&
foldr andb true (map well_formed l₁) wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfl₁: well_formed a = true
∧ foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Pattern.wf l₁
→ Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g b: Pattern Heqb: b = foldr patt_imp g (l₁ ++ h :: l₂) IHl₁: Γ
⊢i (h' ---> h) --->
b ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g b: Pattern Heqb: b = foldr patt_imp g (l₁ ++ h :: l₂) b': Pattern Heqb': b' = foldr patt_imp g (l₁ ++ h' :: l₂) IHl₁: Γ ⊢i (h' ---> h) ---> b ---> b'
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g b: Pattern Heqb: b = foldr patt_imp g (l₁ ++ h :: l₂) b': Pattern Heqb': b' = foldr patt_imp g (l₁ ++ h' :: l₂) IHl₁: Γ ⊢i (h' ---> h) ---> b ---> b'
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ ⊢i (b ---> b') ---> (a ---> b) ---> a ---> b'
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g b: Pattern Heqb: b = foldr patt_imp g (l₁ ++ h :: l₂) b': Pattern Heqb': b' = foldr patt_imp g (l₁ ++ h' :: l₂) IHl₁: Γ ⊢i (h' ---> h) ---> b ---> b'
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂) prf: Γ ⊢i (b ---> b') ---> (a ---> b) ---> a ---> b'
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g b: Pattern Heqb: b = foldr patt_imp g (l₁ ++ h :: l₂) b': Pattern Heqb': b' = foldr patt_imp g (l₁ ++ h' :: l₂) IHl₁: Γ ⊢i (h' ---> h) ---> b ---> b'
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂)
Γ ⊢i (b ---> b') ---> (a ---> b) ---> a ---> b'
using BasicReasoning
apply prf_weaken_conclusion; subst; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g b: Pattern Heqb: b = foldr patt_imp g (l₁ ++ h :: l₂) b': Pattern Heqb': b' = foldr patt_imp g (l₁ ++ h' :: l₂) IHl₁: Γ ⊢i (h' ---> h) ---> b ---> b'
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) g1: Pattern Heqg1: g1 = foldr patt_imp g (h :: l₂) g2: Pattern Heqg2: g2 = foldr patt_imp g (h' :: l₂) prf: Γ ⊢i (b ---> b') ---> (a ---> b) ---> a ---> b'
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed (h' ---> h)
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed
(foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂))
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ ⊢i (h' ---> h) ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i ?B --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i ?B --->
foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂)
using BasicReasoning
apply prf.
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed (h' ---> h)
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed
(foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂))
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed
(foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂))
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
apply IHl₁.
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed (h' ---> h)
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed
(foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂))
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern h, h', g: Pattern wfa: well_formed a = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ wfh: well_formed h wfh': well_formed h' wfg: well_formed g IHl₁: Γ
⊢i (h' ---> h) --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning wfal₁: Pattern.wf (a :: l₁) prf: Γ
⊢i (foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)) --->
(a ---> foldr patt_imp g (l₁ ++ h :: l₂)) --->
a ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
well_formed
(foldr patt_imp g ((a :: l₁) ++ h :: l₂) --->
foldr patt_imp g ((a :: l₁) ++ h' :: l₂))
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo
well_formed A
→ well_formed A'
→ well_formed B
→ Γ ⊢i A' ---> A using i
→ Γ ⊢i (A ---> B) ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo
well_formed A
→ well_formed A'
→ well_formed B
→ Γ ⊢i A' ---> A using i
→ Γ ⊢i (A ---> B) ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i
Γ ⊢i (A ---> B) ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i
Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i H1: Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Γ ⊢i (A ---> B) ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i
Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i
Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using BasicReasoning
apply syllogism; wf_auto2.
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i H1: Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Γ ⊢i (A ---> B) ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i H1: Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i H1: Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Γ ⊢i ?ϕ₁ ---> (A ---> B) ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i H1: Γ ⊢i (A' ---> A) ---> (A ---> B) ---> A' ---> B
using i
Γ ⊢i A' ---> A using i
apply A'impA.Defined.
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo
well_formed A
→ well_formed A'
→ well_formed B
→ Γ ⊢i A' ---> A using i
→ Γ ⊢i A ---> B using i → Γ ⊢i A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo
well_formed A
→ well_formed A'
→ well_formed B
→ Γ ⊢i A' ---> A using i
→ Γ ⊢i A ---> B using i → Γ ⊢i A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
Γ ⊢i A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
Γ ⊢i ?ϕ₁ ---> A' ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
Γ ⊢i ?A ---> B using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed ?A
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed A'
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed B
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
Γ ⊢i A' ---> ?Ausing i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed A
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed A'
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed B
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
Γ ⊢i A' ---> A using i
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed A
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed A'
Σ: Signature Γ: Theory A, A', B: Pattern i: ProofInfo wfA: well_formed A wfA': well_formed A' wfB: well_formed B A'impA: Γ ⊢i A' ---> A using i AimpB: Γ ⊢i A ---> B using i
well_formed B
all: wf_auto2.Defined.
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ
⊢i foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ
⊢i foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i
Γ
⊢i foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g (l₁ ++ h :: l₂) --->
foldr patt_imp g (l₁ ++ h' :: l₂)
using BasicReasoning
apply prf_strenghten_premise_iter; wf_auto2.
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i
Γ ⊢i h' ---> h using i
exact H.Defined.
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂)
using i
→ Γ ⊢i foldr patt_imp g (l₁ ++ h' :: l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂)
using i
→ Γ ⊢i foldr patt_imp g (l₁ ++ h' :: l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Γ ⊢i foldr patt_imp g (l₁ ++ h' :: l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l₁ ++ h' :: l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Γ ⊢i foldr patt_imp g (l₁ ++ ?h :: l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Pattern.wf l₁
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Pattern.wf l₂
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed ?h
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed h'
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed g
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Γ ⊢i h' ---> ?husing i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Pattern.wf l₁
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Pattern.wf l₂
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed h
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed h'
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed g
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Pattern.wf l₁
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
Pattern.wf l₂
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed h
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed h'
Σ: Signature Γ: Theory l₁, l₂: list Pattern h, h', g: Pattern i: ProofInfo WFl₁: Pattern.wf l₁ WFl₂: Pattern.wf l₂ WFh: well_formed h WFh': well_formed h' WFg: well_formed g H: Γ ⊢i h' ---> h using i H0: Γ ⊢i foldr patt_imp g (l₁ ++ h :: l₂) using i
well_formed g
all: wf_auto2.Defined.
Σ: Signature Γ: Theory a, b, c: Pattern
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning
Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning
apply P1; wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning
Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning
Γ ⊢i a ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning H2: Γ ⊢i a ---> c ---> a using BasicReasoning
Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning
Γ ⊢i a ---> c ---> a using BasicReasoning
apply P1; wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning H2: Γ ⊢i a ---> c ---> a using BasicReasoning
Γ ⊢i a ---> b ---> c ---> a using BasicReasoning
eapply (@syllogism_meta _ _ _ _ _ _ _ _ _ H2 H1).
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning H2: Γ ⊢i a ---> c ---> a using BasicReasoning
well_formed a
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning H2: Γ ⊢i a ---> c ---> a using BasicReasoning
well_formed (c ---> a)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (c ---> a) ---> b ---> c ---> a
using BasicReasoning H2: Γ ⊢i a ---> c ---> a using BasicReasoning
well_formed (b ---> c ---> a)
all: wf_auto2.Defined.(* This will form a base for the tactic 'exact 0' *)
Σ: Signature Γ: Theory a: Pattern l: list Pattern
well_formed a
→ Pattern.wf l
→ Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern
well_formed a
→ Pattern.wf l
→ Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern wfa: well_formed a wfl: Pattern.wf l
Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a wfl: Pattern.wf []
Γ ⊢i a ---> a using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l → Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a wfl: Pattern.wf []
Γ ⊢i a ---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a wfl: Pattern.wf []
well_formed a
exact wfa.
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l → Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l → Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfl: foldr andb true (map well_formed (a0 :: l)) IHl: Pattern.wf l → Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfl: well_formed a0 && foldr andb true (map well_formed l) IHl: Pattern.wf l → Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfl: well_formed a0 = true ∧ foldr andb true (map well_formed l) = true IHl: Pattern.wf l → Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Pattern.wf l
→ Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp a l ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp a l ---> a0 ---> foldr patt_imp a l
using BasicReasoning
apply P1; wf_auto2.
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
Γ ⊢i a ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed a
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed (a0 ---> foldr patt_imp a l)
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
Γ ⊢i a ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
Γ ⊢i ?B ---> a0 ---> foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed a
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed (foldr patt_imp a l)
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed (a0 ---> foldr patt_imp a l)
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
Γ ⊢i a ---> foldr patt_imp a l using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed a
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed (foldr patt_imp a l)
Σ: Signature Γ: Theory a, a0: Pattern l: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true IHl: Γ ⊢i a ---> foldr patt_imp a l
using BasicReasoning wfa0l: Pattern.wf (a0 :: l) H1: Γ
⊢i foldr patt_imp a l --->
a0 ---> foldr patt_imp a l
using BasicReasoning
well_formed (a0 ---> foldr patt_imp a l)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern
well_formed a
→ Pattern.wf l₁
→ Pattern.wf l₂
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern
well_formed a
→ Pattern.wf l₁
→ Pattern.wf l₂
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: Pattern.wf l₁ wfl₂: Pattern.wf l₂
Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern wfa: well_formed a wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂
Γ ⊢i a ---> foldr patt_imp a l₂ using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: Pattern.wf (a0 :: l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern wfa: well_formed a wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂
Γ ⊢i a ---> foldr patt_imp a l₂ using BasicReasoning
apply nested_const; wf_auto2.
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: Pattern.wf (a0 :: l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: Pattern.wf (a0 :: l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: foldr andb true (map well_formed (a0 :: l₁)) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: well_formed a0 &&
foldr andb true (map well_formed l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfl₁: well_formed a0 = true
∧ foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i ?ϕ₁ ---> a0 ---> foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed (foldr patt_imp a (l₁ ++ a :: l₂))
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed a0
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed (foldr patt_imp a (l₁ ++ a :: l₂))
Σ: Signature Γ: Theory a, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfa0: well_formed a0 = true wfl₁: foldr andb true (map well_formed l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ ⊢i foldr patt_imp a (l₁ ++ a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed a0
all: wf_auto2.Defined.
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern
well_formed a
→ well_formed b
→ well_formed g
→ Pattern.wf l₁
→ Pattern.wf l₂
→ Γ
⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) --->
foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern
well_formed a
→ well_formed b
→ well_formed g
→ Pattern.wf l₁
→ Pattern.wf l₂
→ Γ
⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) --->
foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: Pattern.wf l₁ wfl₂: Pattern.wf l₂
Γ
⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) --->
foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g: Pattern l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂
Γ
⊢i (a ---> b ---> foldr patt_imp g l₂) --->
b ---> a ---> foldr patt_imp g l₂
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: Pattern.wf (a0 :: l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g: Pattern l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: Pattern.wf [] wfl₂: Pattern.wf l₂
Γ
⊢i (a ---> b ---> foldr patt_imp g l₂) --->
b ---> a ---> foldr patt_imp g l₂
using BasicReasoning
apply reorder; wf_auto2.
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: Pattern.wf (a0 :: l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: Pattern.wf (a0 :: l₁) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: foldr andb true (map well_formed (a0 :: l₁)) wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfl₁: well_formed a0 = true
∧ (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Pattern.wf l₁
→ Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ
⊢i (a0 ---> foldr patt_imp g (l₁ ++ a :: b :: l₂)) --->
a0 ---> foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed a0
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed (foldr patt_imp g (l₁ ++ a :: b :: l₂))
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed (foldr patt_imp g (l₁ ++ b :: a :: l₂))
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed a0
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed (foldr patt_imp g (l₁ ++ a :: b :: l₂))
Σ: Signature Γ: Theory a, b, g, a0: Pattern l₁, l₂: list Pattern wfa: well_formed a wfb: well_formed b wfg: well_formed g wfa0: well_formed a0 = true wfl₁: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l₁) = true wfl₂: Pattern.wf l₂ IHl₁: Γ
⊢i foldr patt_imp g (l₁ ++ a :: b :: l₂) --->
foldr patt_imp g (l₁ ++ b :: a :: l₂)
using BasicReasoning wfa0l₁: Pattern.wf (a0 :: l₁)
well_formed (foldr patt_imp g (l₁ ++ b :: a :: l₂))
all: wf_auto2.Defined.
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed g
→ Pattern.wf l₁
→ Pattern.wf l₂
→ Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂)
using i
→ Γ ⊢i foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using i
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed g
→ Pattern.wf l₁
→ Pattern.wf l₂
→ Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂)
using i
→ Γ ⊢i foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using i
(* TODO we should have a function/lemma for creating these "meta" variants. *)
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFg: well_formed g Wfl1: Pattern.wf l₁ Wfl2: Pattern.wf l₂ H: Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
Γ ⊢i foldr patt_imp g (l₁ ++ [b; a] ++ l₂) using i
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFg: well_formed g Wfl1: Pattern.wf l₁ Wfl2: Pattern.wf l₂ H: Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFg: well_formed g Wfl1: Pattern.wf l₁ Wfl2: Pattern.wf l₂ H: Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using i
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFg: well_formed g Wfl1: Pattern.wf l₁ Wfl2: Pattern.wf l₂ H: Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using i
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFg: well_formed g Wfl1: Pattern.wf l₁ Wfl2: Pattern.wf l₂ H: Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l₁ ++ [b; a] ++ l₂)
using BasicReasoning
apply prf_reorder_iter; wf_auto2.
Σ: Signature Γ: Theory a, b, g: Pattern l₁, l₂: list Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFg: well_formed g Wfl1: Pattern.wf l₁ Wfl2: Pattern.wf l₂ H: Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
Γ ⊢i foldr patt_imp g (l₁ ++ [a; b] ++ l₂) using i
exact H.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A ---> ! ! B using i → Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A ---> ! ! B using i → Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A ---> ! ! B using i
Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A ---> ! ! B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A ---> ! ! B using i
Γ ⊢i ?ϕ₁ ---> A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A ---> ! ! B using i
Γ ⊢i ?ϕ₁ ---> A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A ---> ! ! B using i
Γ ⊢i ?ϕ₁ ---> A ---> B using BasicReasoning
apply A_impl_not_not_B; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A ---> ! ! B using i
Γ ⊢i A ---> ! ! B using i
exact H.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A and B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A and B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A and B ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! (! A or ! B) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
apply disj_left_intro; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
apply modus_ponens; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ((! A or ! B ---> ⊥) ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! A ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ?B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A or ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ((! A or ! B ---> ⊥) ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! A ---> ! A or ! B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A or ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ((! A or ! B ---> ⊥) ---> ⊥)
all: wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H2: Γ ⊢i (! A or ! B ---> ⊥) ---> ! A ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H2: Γ ⊢i (! A or ! B ---> ⊥) ---> ! A ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> ! ! A
using BasicReasoning
apply H2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A or ! B ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! A ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! A ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ⊥
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A and B ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A and B ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A and B ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! (! A or ! B) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
apply disj_right_intro; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
apply modus_ponens; wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ((! A or ! B ---> ⊥) ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! B ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ?B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A or ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ((! A or ! B ---> ⊥) ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i ! B ---> ! A or ! B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A or ! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ((! A or ! B ---> ⊥) ---> ⊥)
all: wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H2: Γ ⊢i (! A or ! B ---> ⊥) ---> ! B ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H2: Γ ⊢i (! A or ! B ---> ⊥) ---> ! B ---> ⊥
using BasicReasoning
Γ ⊢i (! A or ! B ---> ⊥) ---> ! ! B
using BasicReasoning
apply H2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! B)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed (! A or ! B ---> ⊥)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B H: Γ ⊢i ! B ---> ! A or ! B using BasicReasoning H0: Γ ⊢i ! A or ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning H1: Γ ⊢i ! B ---> (! A or ! B ---> ⊥) ---> ⊥
using BasicReasoning
well_formed ⊥
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A and B using i → Γ ⊢i A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A and B using i → Γ ⊢i A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?ϕ₁ ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?ϕ₁ ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?ϕ₁ ---> A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed A
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed ?B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed ?B
shelve.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i A and?Busing i
1: apply H.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed B
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A and B using i → Γ ⊢i B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A and B using i → Γ ⊢i B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?ϕ₁ ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i ?Aand B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed ?A
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed A
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed B
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A or ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A or ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
Γ ⊢i A or ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
Γ ⊢i ! A ---> ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
well_formed (! A)
wf_auto2.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B using i
→ Γ ⊢i A ---> ! B using i → Γ ⊢i ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B using i
→ Γ ⊢i A ---> ! B using i → Γ ⊢i ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i
Γ ⊢i ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i (A ---> ! B) ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning H2: Γ ⊢i (A ---> ! B) ---> ! A using i
Γ ⊢i ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i (A ---> ! B) ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (A ---> ! B) ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (A ---> ! B) ---> ! A using i
apply useBasicReasoning; apply H1.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning
Γ ⊢i A ---> B using i
exact AimpB.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning H2: Γ ⊢i (A ---> ! B) ---> ! A using i
Γ ⊢i ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning H2: Γ ⊢i (A ---> ! B) ---> ! A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning H2: Γ ⊢i (A ---> ! B) ---> ! A using i
Γ ⊢i ?ϕ₁ ---> ! A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning H2: Γ ⊢i (A ---> ! B) ---> ! A using i
Γ ⊢i ?ϕ₁ ---> ! A using i
apply H2.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimpB: Γ ⊢i A ---> B using i AimpnB: Γ ⊢i A ---> ! B using i H1: Γ ⊢i (A ---> B) ---> (A ---> ! B) ---> ! A
using BasicReasoning H2: Γ ⊢i (A ---> ! B) ---> ! A using i
Γ ⊢i g ---> g' using i
→ Γ Ⱶ
l--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g'
using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: Γ Ⱶ
l--------------------------------------
g
using i
Γ Ⱶ
l--------------------------------------
g'
using i
(*mlExtractWF wfl wfgp.*)
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g'
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g'
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
g'
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g'
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g'
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g'
using i
)
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i
well_formed g'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g' (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g' (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed (g ---> g')
Γ ⊢i foldr patt_imp g' (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Γ ⊢i foldr patt_imp g' (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed ?g
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g'
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Γ ⊢i ?g ---> g' using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Γ ⊢i foldr patt_imp ?g (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g'
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Γ ⊢i g ---> g' using i
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g'
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
well_formed g
Σ: Signature Γ: Theory l: hypotheses g, g': Pattern i: ProofInfo Hgg': Γ ⊢i g ---> g' using i Hlg: well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l)
using i wfg': well_formed g' wfl: Pattern.wf (patterns_of l) wfimp: well_formed g
Pattern.wf (patterns_of l)
all: assumption.Defined.
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i a ---> (a ---> b) ---> b using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning
Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i a ---> (a ---> b) ---> b using BasicReasoning
apply modus_ponens; assumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning
Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning
Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning H2: Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning
Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
apply P2; wf_auto2.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning H2: Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
Γ ⊢i (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning H2: Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning H2: Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (a ---> a ---> b) ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b H1: Γ ⊢i a ---> (a ---> b) ---> b
using BasicReasoning H2: Γ
⊢i (a ---> (a ---> b) ---> b) --->
(a ---> a ---> b) ---> a ---> b
using BasicReasoning
Γ ⊢i a ---> (a ---> b) ---> b using BasicReasoning
apply H1.Defined.
Σ: Signature Γ: Theory a, b, c: Pattern
well_formed a
→ well_formed b
→ well_formed c
→ Γ
⊢i (a ---> b) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern
well_formed a
→ well_formed b
→ well_formed c
→ Γ
⊢i (a ---> b) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i (a ---> b) ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
apply reorder; wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning
apply H1.
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed (a ---> b)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed ((a ---> b ---> c) ---> a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i ?B ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed (a ---> b)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed ((b ---> a ---> c) ---> a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed ((a ---> b ---> c) ---> a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i (a ---> b ---> c) ---> b ---> a ---> c
using BasicReasoning H2: Γ
⊢i ((b ---> a ---> c) ---> a ---> c) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c
Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning
apply syllogism; wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning
apply prf_contraction; wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Γ ⊢i (a ---> b) ---> (b ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Γ
⊢i foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c] using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Pattern.wf [a ---> b; b ---> a ---> c]
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
well_formed ?g
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
well_formed (a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Γ ⊢i ?g ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Γ ⊢i foldr patt_imp ?g [a ---> b; b ---> a ---> c]
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Pattern.wf [a ---> b; b ---> a ---> c]
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
well_formed (a ---> a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
well_formed (a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
Pattern.wf [a ---> b; b ---> a ---> c]
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
well_formed (a ---> a ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern wfa: well_formed a wfb: well_formed b wfc: well_formed c H3: Γ
⊢i (a ---> b) --->
(b ---> a ---> c) ---> a ---> a ---> c
using BasicReasoning H4: Γ ⊢i (a ---> a ---> c) ---> a ---> c
using BasicReasoning Hiter: ((a ---> b) --->
(b ---> a ---> c) ---> a ---> c) =
foldr patt_imp (a ---> c)
[a ---> b; b ---> a ---> c]
well_formed (a ---> c)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b using i
→ Γ ⊢i (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b using i
→ Γ ⊢i (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b using i
Γ ⊢i (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b using i
Γ ⊢i ?ϕ₁ ---> (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b using i
Γ ⊢i ?ϕ₁ ---> (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b using i
Γ ⊢i ?ϕ₁ ---> (a ---> b ---> c) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b using i
Γ ⊢i a ---> b using i
exact H.Defined.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b using i
→ Γ ⊢i a ---> b ---> c using i
→ Γ ⊢i a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b using i
→ Γ ⊢i a ---> b ---> c using i
→ Γ ⊢i a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i ?ϕ₁ ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i ?ϕ₁using i
apply H2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i (a ---> b ---> c) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
well_formed a
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
well_formed b
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
well_formed c
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
Γ ⊢i a ---> b using i
apply H1.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
well_formed a
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
well_formed b
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H1: Γ ⊢i a ---> b using i H2: Γ ⊢i a ---> b ---> c using i
well_formed c
all: wf_auto2.}Defined.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ
⊢i ((g ---> g') ---> foldr patt_imp g l) --->
(g ---> g') ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ
⊢i ((g ---> g') ---> foldr patt_imp g l) --->
(g ---> g') ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g'
Γ
⊢i ((g ---> g') ---> foldr patt_imp g l) --->
(g ---> g') ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H1: Γ
⊢i (g ---> g') --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Γ
⊢i ((g ---> g') ---> foldr patt_imp g l) --->
(g ---> g') ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') H1: Γ
⊢i a --->
foldr patt_imp g l ---> foldr patt_imp g' l
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g' l using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l H1: Γ ⊢i a ---> b ---> foldr patt_imp g' l
using BasicReasoning
Γ ⊢i (a ---> b) ---> a ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b ---> c) --->
(a ---> b) ---> a ---> c using BasicReasoning
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed (a ---> b)
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed (a ---> b ---> c)
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b) --->
(a ---> b ---> c) ---> a ---> c
using BasicReasoning
well_formed (a ---> c)
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b ---> c) --->
(a ---> b) ---> a ---> c using BasicReasoning
Γ ⊢i (a ---> b) ---> a ---> c using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b ---> c) --->
(a ---> b) ---> a ---> c using BasicReasoning
Γ ⊢i ?ϕ₁using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b ---> c) --->
(a ---> b) ---> a ---> c using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (a ---> b) ---> a ---> c
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' a: Pattern Heqa: a = (g ---> g') b: Pattern Heqb: b = foldr patt_imp g l c: Pattern Heqc: c = foldr patt_imp g' l H1: Γ ⊢i a ---> b ---> c using BasicReasoning H: well_formed a H0: well_formed b H2: well_formed c H2': Γ
⊢i (a ---> b ---> c) --->
(a ---> b) ---> a ---> c using BasicReasoning
Γ ⊢i a ---> b ---> c using BasicReasoning
apply H1.Defined.
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
→ Γ ⊢i (g ---> g') ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed g'
→ Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
→ Γ ⊢i (g ---> g') ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i (g ---> g') ---> foldr patt_imp g' l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i ?ϕ₁ ---> (g ---> g') ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i ?ϕ₁ ---> (g ---> g') ---> foldr patt_imp g' l
using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i ?ϕ₁ ---> (g ---> g') ---> foldr patt_imp g' l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, g': Pattern i: ProofInfo wfl: Pattern.wf l wfg: well_formed g wfg': well_formed g' H: Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
Γ ⊢i (g ---> g') ---> foldr patt_imp g l using i
exact H.}Defined.
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, g': Pattern i: ProofInfo
Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
→ Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, g': Pattern i: ProofInfo
Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
→ Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed
(mlConclusion
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
)))
→ mlTheory
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
)))
using mlInfo
(Γ Ⱶ
(name ∶ g ---> g') :: l--------------------------------------
g'
using i
)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf ((g ---> g') :: patterns_of l)
→ Γ
⊢i (g ---> g') --->
foldr patt_imp g (patterns_of l) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo
Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
→ Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo
Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
→ Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo
(well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g
using i
))
→ well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (name ∶ g ---> g') :: l₂--------------------------------------
g'
using i
)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo
(well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂)) using i)
→ well_formed g'
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g'
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g'
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt
(l₁ ++ (name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g'
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g'
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g'
(map nh_patt (l₁ ++ (name ∶ g ---> g') :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g'
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂)) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Pattern.wf (map nh_patt l₁)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g'
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g'
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g'
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g'
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
well_formed g'
exact wfg'.
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(patterns_of (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo H: well_formed g
→ Pattern.wf
(map nh_patt (l₁ ++ (name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt
(l₁ ++ (name ∶ g ---> g') :: l₂))
using i wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂)) H: well_formed g
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
using i
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
(g ---> g')
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂)) H: well_formed g
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
using i
well_formed g
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis name: string g, g': Pattern i: ProofInfo wfg': well_formed g' wfl₁gg'l₂: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂)) H: well_formed g
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((name ∶ g ---> g') :: l₂))
using i
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
(p ---> q) --->
(r ---> s) ---> (! p ---> r) ---> ! q ---> s
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
--------------------------------------
(r ---> s) ---> (! p ---> r) ---> ! q ---> s
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
--------------------------------------
(! p ---> r) ---> ! q ---> s
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
--------------------------------------
! q ---> s
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
"H3" ∶ ! q,
--------------------------------------
s
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
"H3" ∶ ! q,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
"H3" ∶ ! q,
--------------------------------------
! p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
"H3" ∶ ! q,
"H4" ∶ p,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
"H3" ∶ q ---> ⊥,
"H4" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r, s: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r wfs: well_formed s
Γ Ⱶ
"H0" ∶ p ---> q,
"H1" ∶ r ---> s,
"H2" ∶ ! p ---> r,
"H3" ∶ q ---> ⊥,
"H4" ∶ p,
--------------------------------------
p
using BasicReasoning
mlExact "H4".Defined.
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Γ ⊢i b using i → Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Γ ⊢i b using i → Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i ?ϕ₁ ---> a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i b ---> a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i b ---> a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b H: Γ ⊢i b using i
Γ ⊢i b ---> a ---> b using BasicReasoning
apply P1; wf_auto2.}Defined.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b ---> c using i
→ Γ ⊢i (a ---> b) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b ---> c using i
→ Γ ⊢i (a ---> b) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i (a ---> b) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i ?ϕ₁ ---> (a ---> b) ---> a ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i (a ---> b ---> c) ---> (a ---> b) ---> a ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i (a ---> b ---> c) ---> (a ---> b) ---> a ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i (a ---> b ---> c) ---> (a ---> b) ---> a ---> c
using BasicReasoning
apply P2; wf_auto2.}Defined.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern wfl: Pattern.wf l wfg: well_formed g wfh: well_formed h
Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory g, h: Pattern wfl: Pattern.wf [] wfg: well_formed g wfh: well_formed h
Γ ⊢i h ---> (h ---> g) ---> g using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfh: well_formed h IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory g, h: Pattern wfl: Pattern.wf [] wfg: well_formed g wfh: well_formed h
Γ ⊢i h ---> (h ---> g) ---> g using BasicReasoning
apply modus_ponens; auto.
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfh: well_formed h IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfh: well_formed h IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfl: foldr andb true (map well_formed (a :: l)) wfg: well_formed g wfh: well_formed h IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfl: well_formed a = true
∧ (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
apply prf_add_assumption; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
apply H1.
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning
Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
apply P2; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed (a ---> foldr patt_imp h l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed
((a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Γ
⊢i ?B --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Γ ⊢i (a ---> foldr patt_imp h l) ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed (a ---> foldr patt_imp h l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed
(a --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed
((a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed (a ---> foldr patt_imp h l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed
(a --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a: Pattern l: list Pattern g, h: Pattern wfa: well_formed a = true wfl: (fix fold_right (l : list bool) : bool :=
match l with
| [] => true
| b :: t => b && fold_right t
end)
((fix map (l : list Pattern) : list bool :=
match l with
| [] => []
| a :: t => well_formed a :: map t
end) l) = true wfg: well_formed g wfh: well_formed h IHl: Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning wfal: Pattern.wf (a :: l) H1: Γ
⊢i a --->
foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l) --->
a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l) --->
(a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l using BasicReasoning
well_formed
((a ---> foldr patt_imp g (l ++ [h])) --->
a ---> foldr patt_imp g l)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l using i
→ Γ
⊢i foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l using i
→ Γ
⊢i foldr patt_imp g (l ++ [h]) --->
foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ
⊢i foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Γ
⊢i foldr patt_imp h l --->
foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
Pattern.wf l
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
well_formed g
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i
well_formed h
all: wf_auto2.}Defined.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l using i
→ Γ ⊢i foldr patt_imp g (l ++ [h]) using i
→ Γ ⊢i foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l using i
→ Γ ⊢i foldr patt_imp g (l ++ [h]) using i
→ Γ ⊢i foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ ⊢i foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ ⊢i ?ϕ₁using i
apply H0.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ
⊢i foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ
⊢i foldr patt_imp g (l ++ [h]) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Pattern.wf l
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
well_formed g
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
well_formed h
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Γ ⊢i foldr patt_imp h l using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
Pattern.wf l
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
well_formed g
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l using i H0: Γ ⊢i foldr patt_imp g (l ++ [h]) using i
well_formed h
→ Γ Ⱶ
l--------------------------------------
h
using i
→ Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g
using i
well_formed h
→ Γ Ⱶ
l--------------------------------------
h
using i
→ Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: Γ Ⱶ
l--------------------------------------
h
using i
H2: Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
h
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
h
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
h
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
h
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
h
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
h
using i
) H2: well_formed
(mlConclusion
(Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l ++ (name ∶ h,
)--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i
)
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i
well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed g
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed ?h
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp ?h (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (patterns_of l ++ [?h]) using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed g
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (patterns_of l ++ [h]) using i
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed g
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i H2: well_formed g
→ Pattern.wf (patterns_of (l ++ (name ∶ h,
)))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l ++ (name ∶ h,
))) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l) H2: well_formed g
→ Pattern.wf
(map nh_patt l ++ map nh_patt (name ∶ h,
))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l ++
map nh_patt (name ∶ h,
)) using i
well_formed g
Σ: Signature Γ: Theory l: hypotheses name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp h (patterns_of l) using i wfg: well_formed g wfl: Pattern.wf (patterns_of l) H2: well_formed g
→ Pattern.wf
(map nh_patt l ++ map nh_patt (name ∶ h,
))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l ++
map nh_patt (name ∶ h,
)) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf l1 wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h
Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
Σ: Signature Γ: Theory l2: list Pattern g, h: Pattern wfl1: Pattern.wf [] wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h
Γ
⊢i h --->
(h ---> foldr patt_imp g l2) --->
foldr patt_imp g l2 using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf (a :: l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory l2: list Pattern g, h: Pattern wfl1: Pattern.wf [] wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h
Γ
⊢i h --->
(h ---> foldr patt_imp g l2) --->
foldr patt_imp g l2 using BasicReasoning
apply modus_ponens; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf (a :: l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf (a :: l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: foldr andb true (map well_formed (a :: l1)) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: well_formed a &&
foldr andb true (map well_formed l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: well_formed a = true
∧ foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1)
Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
apply prf_add_assumption; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
exact H1.
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
apply P2; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed (a ---> foldr patt_imp h l1)
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed
((a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2))
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i ?B --->
(a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ ⊢i (a ---> foldr patt_imp h l1) ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed (a ---> foldr patt_imp h l1)
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed
(a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2))
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed
((a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2))
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed (a ---> foldr patt_imp h l1)
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed
(a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2))
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning wfal1: Pattern.wf (a :: l1) H1: Γ
⊢i a --->
foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H2: Γ
⊢i (a ---> foldr patt_imp h l1) --->
a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)
using BasicReasoning H3: Γ
⊢i (a --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2)) --->
(a ---> foldr patt_imp g (l1 ++ [h] ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ l2)
using BasicReasoning
well_formed
((a ---> foldr patt_imp g (l1 ++ h :: l2)) --->
a ---> foldr patt_imp g (l1 ++ l2))
all: wf_auto2.Defined.
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l1 using i
→ Γ
⊢i foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l1 using i
→ Γ
⊢i foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ
⊢i foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ ⊢i ?ϕ₁using i
apply H.
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i
Γ
⊢i foldr patt_imp h l1 --->
foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using BasicReasoning
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l1 using i
→ Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2)
using i
→ Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp h l1 using i
→ Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2)
using i
→ Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ ⊢i ?ϕ₁using i
apply H0.
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ
⊢i foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ
⊢i foldr patt_imp g (l1 ++ [h] ++ l2) --->
foldr patt_imp g (l1 ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Pattern.wf l1
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Pattern.wf l2
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
well_formed g
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
well_formed h
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Γ ⊢i foldr patt_imp h l1 using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Pattern.wf l1
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Pattern.wf l2
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
well_formed g
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo WFl1: Pattern.wf l1 WFl2: Pattern.wf l2 WFg: well_formed g WGh: well_formed h H: Γ ⊢i foldr patt_imp h l1 using i H0: Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
well_formed h
all: wf_auto2.}Defined.
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo
well_formed h
→ Γ Ⱶ
l1--------------------------------------
h
using i
→ Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
→ Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo
well_formed h
→ Γ Ⱶ
l1--------------------------------------
h
using i
→ Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
→ Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: Γ Ⱶ
l1--------------------------------------
h
using i
H2: Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed
(mlConclusion
(Γ Ⱶ
l1--------------------------------------
h
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1--------------------------------------
h
using i
)))
→ mlTheory
(Γ Ⱶ
l1--------------------------------------
h
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l1--------------------------------------
h
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1--------------------------------------
h
using i
)))
using mlInfo
(Γ Ⱶ
l1--------------------------------------
h
using i
) H2: well_formed
(mlConclusion
(Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l1 ++ (name ∶ h,
) ++ l2--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i
well_formed g
→ Pattern.wf (patterns_of (l1 ++ l2))
→ Γ ⊢i foldr patt_imp g (patterns_of (l1 ++ l2))
using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Γ ⊢i foldr patt_imp g (patterns_of (l1 ++ l2)) using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Γ ⊢i foldr patt_imp g (map nh_patt (l1 ++ l2)) using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Γ
⊢i foldr patt_imp g (map nh_patt l1 ++ map nh_patt l2)
using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (map nh_patt l2)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed g
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed ?h
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Γ ⊢i foldr patt_imp ?h (map nh_patt l1) using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ [?h] ++ map nh_patt l2)
using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (map nh_patt l2)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed g
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed h
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed h
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (patterns_of l1)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ [h] ++ map nh_patt l2) using i
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (map nh_patt l2)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed g
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed h
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
well_formed h
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i H2: well_formed g
→ Pattern.wf
(patterns_of (l1 ++ (name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l1 ++ (name ∶ h) :: l2))
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2))
Pattern.wf (patterns_of l1)
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2)) H2: well_formed g
→ Pattern.wf
(map nh_patt l1 ++
map nh_patt ((name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++
map nh_patt ((name ∶ h) :: l2)) using i
well_formed g
Σ: Signature Γ: Theory l1: hypotheses l2: list named_hypothesis name: string g, h: Pattern i: ProofInfo wfh: well_formed h H1: well_formed h
→ Pattern.wf (patterns_of l1)
→ Γ ⊢i foldr patt_imp h (patterns_of l1)
using i wfg: well_formed g wfl1l2: Pattern.wf (patterns_of (l1 ++ l2)) H2: well_formed g
→ Pattern.wf
(map nh_patt l1 ++
map nh_patt ((name ∶ h) :: l2))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++
map nh_patt ((name ∶ h) :: l2)) using i
abstract(
pose proof (wfl1 := wf_take (length (patterns_of l1)) wfl1l2);
unfold patterns_of in wfl1;
rewrite map_app in wfl1;
rewrite take_app in wfl1;
pose proof (wfl2 := wf_drop (length (patterns_of l1)) wfl1l2);
unfold patterns_of in wfl2;
rewrite map_app in wfl2;
rewrite drop_app in wfl2;
unfold Pattern.wf; rewrite map_app; rewrite foldr_app;
simpl; rewrite wfh; unfold Pattern.wfin wfl2; rewrite wfl2;
simpl; exact wfl1
).}Defined.Tactic Notation"_mlAssert_nocheck""("constr(name) ":"constr(t) ")" :=
match goal with
| |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm?Ctx?l?g?i) =>
letHwf := fresh"Hwf"inassert (Hwf : well_formed t);
[idtac|
letH := fresh"H"inassert (H : @mkMLGoal Sgm Ctx l t i);
[ | (eapply (@mlGoal_assert Sgm Ctx l name g t i Hwf H); rewrite [_ ++ _]/=; clear H)]
]
end.(* TODO: make this bind tigther. *)Tactic Notation"mlAssert""("constr(name) ":"constr(t) ")" :=
_failIfUsed name;
_mlAssert_nocheck (name : t)
.Tactic Notation"mlAssert""("constr (t) ")" :=
lethyps := _getHypNames inletname := evallazyin (fresh hyps) in
mlAssert (name : (t)).
Σ: Signature Γ: Theory a: Pattern
well_formed a
→ Γ ⊢i a ---> a ---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern
well_formed a
→ Γ ⊢i a ---> a ---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ ⊢i a ---> a ---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
well_formed (a ---> a ---> a)
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
a ---> a ---> a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
well_formed (a ---> a ---> a)
wf_auto2.
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
a ---> a ---> a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H0" ∶ a,
--------------------------------------
a ---> a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
well_formed a
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
well_formed a
wf_auto2.
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
--------------------------------------
a
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
well_formed a
Σ: Signature Γ: Theory a: Pattern wfa, Hwf, Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf, Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
"0" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf: well_formed a
well_formed a
wf_auto2.
Σ: Signature Γ: Theory a: Pattern wfa, Hwf, Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf, Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
"0" ∶ a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa, Hwf, Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory a: Pattern wfa, Hwf, Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ a,
"H2" ∶ a,
"0" ∶ a,
--------------------------------------
a
using BasicReasoning
mlExact "H2".Qed.Ltac_getGoalProofInfo :=
lazymatch goal with
| |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm?Ctx?l?g?i)
=> i
end.Ltac_getGoalTheory :=
lazymatch goal with
| |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm?Ctx?l?g?i)
=> Ctx
end.Tactic Notation"mlAssert""("constr(name) ":"constr(t) ")""using""first"constr(n) :=
_failIfUsed name;
lazymatch goal with
| |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm?Ctx?l?g?i) =>
letl1 := fresh"l1"inletl2 := fresh"l2"inletHeql1 := fresh"Heql1"inletHeql2 := fresh"Heql2"inremember (take n l) as l1 eqn:Heql1 in |-;
remember (drop n l) as l2 eqn:Heql2 in |-;
simplin Heql1; simplin Heql2;
eapply cast_proof_ml_hyps;
[(
rewrite -[l](take_drop n);
reflexivity
)|];
letHwf := fresh"Hwf"inassert (Hwf : well_formed t);
[idtac|
letH := fresh"H"inassert (H : @mkMLGoal Sgm Ctx l1 t i) ;
[
(eapply cast_proof_ml_hyps; [(rewrite Heql1; reflexivity)|]); clear l1 l2 Heql1 Heql2
| apply (cast_proof_ml_hyps (f_equal patterns_of Heql1)) in H;
eapply (@mlGoal_assert_generalized Sgm Ctx (take n l) (drop n l) name g t i Hwf H);
rewrite [_ ++ _]/=; clear l1 l2 Heql1 Heql2 H]
]
end.Tactic Notation"mlAssert""("constr(t) ")""using""first"constr(n) :=
lethyps := _getHypNames inletname := evalcbvin (fresh hyps) in
mlAssert (name : t) usingfirst n.
Σ: Signature Γ: Theory p, q, a, b: Pattern
well_formed a = true
→ well_formed b = true
→ well_formed p = true
→ well_formed q = true
→ Γ ⊢i a ---> p and q ---> b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern
well_formed a = true
→ well_formed b = true
→ well_formed p = true
→ well_formed q = true
→ Γ ⊢i a ---> p and q ---> b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
Γ ⊢i a ---> p and q ---> b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
well_formed (a ---> p and q ---> b ---> ! ! q)
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
Γ Ⱶ
a ---> p and q ---> b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
well_formed (a ---> p and q ---> b ---> ! ! q)
wf_auto2.
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
Γ Ⱶ
a ---> p and q ---> b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
Γ Ⱶ
"H0" ∶ a,
--------------------------------------
p and q ---> b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
b ---> ! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true l1: list named_hypothesis Heql1: l1 = ("H0" ∶ a,
"H1" ∶ p and q,
) l2: list named_hypothesis Heql2: l2 = ("H2" ∶ b,
)
well_formed p
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"H4" ∶ p,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true l1: list named_hypothesis Heql1: l1 = ("H0" ∶ a,
"H1" ∶ p and q,
) l2: list named_hypothesis Heql2: l2 = ("H2" ∶ b,
)
well_formed p
wf_auto2.
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"H4" ∶ p,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
admit.
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"H4" ∶ p,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p l1: list named_hypothesis Heql1: l1 = ("H0" ∶ a,
"H1" ∶ p and q,
) l2: list named_hypothesis Heql2: l2 = ("H4" ∶ p,
"H2" ∶ b,
)
well_formed p
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf, Hwf0: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf, Hwf0: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"0" ∶ p,
"H4" ∶ p,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf: well_formed p l1: list named_hypothesis Heql1: l1 = ("H0" ∶ a,
"H1" ∶ p and q,
) l2: list named_hypothesis Heql2: l2 = ("H4" ∶ p,
"H2" ∶ b,
)
well_formed p
wf_auto2.
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf, Hwf0: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf, Hwf0: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"0" ∶ p,
"H4" ∶ p,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf, Hwf0: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
admit.
Σ: Signature Γ: Theory p, q, a, b: Pattern wfa: well_formed a = true wfb: well_formed b = true wfp: well_formed p = true wfq: well_formed q = true Hwf, Hwf0: well_formed p
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p and q,
"0" ∶ p,
"H4" ∶ p,
"H2" ∶ b,
--------------------------------------
! ! q
using BasicReasoning
Abort.
Σ: Signature Γ: Theory A: Pattern
well_formed A
→ Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A
→ Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
Γ ⊢i (! A ---> ! ! A) ---> ! ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
Γ ⊢i (! A ---> ! ! A) ---> ! ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A
well_formed (! A)
wf_auto2.
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed (! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
Γ ⊢i A ---> ! ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed (! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed (! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning
well_formed A
all: wf_auto2.
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed ?A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> A) ---> ?Ausing BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i ?A ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> ! ! A) ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i ?B ---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> ! ! A) ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> ! ! A) ---> ! ! A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
Γ ⊢i (! A ---> ! ! A) ---> ! ! A using BasicReasoning
apply H1.
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! A ---> ! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed (! ! A)
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
well_formed A
Σ: Signature Γ: Theory A: Pattern wfA: well_formed A H1: Γ ⊢i (! A ---> ! ! A) ---> ! ! A
using BasicReasoning H2: Γ ⊢i (! A ---> A) ---> ! A ---> ! ! A
using BasicReasoning
foldr patt_imp (p ---> q) l =
foldr patt_imp (foldr patt_imp q [p]) l
reflexivity.Defined.
Σ: Signature Γ: Theory p, q, r: Pattern
well_formed p
→ well_formed q
→ well_formed r
→ Γ
⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern
well_formed p
→ well_formed q
→ well_formed r
→ Γ
⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning
Γ ⊢i r or r ---> r using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning H: Γ ⊢i r or r ---> r using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning
Γ ⊢i r or r ---> r using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning
Γ ⊢i (! r ---> r) ---> r using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning
well_formed r
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning H: Γ ⊢i r or r ---> r using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ ⊢i ?ψusing BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning H: Γ ⊢i r or r ---> r using BasicReasoning
?ψ =
((p ---> r) ---> (q ---> r) ---> p or q ---> r or r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning H: Γ ⊢i r or r ---> r using BasicReasoning
?ψ =
((p ---> r) ---> (q ---> r) ---> p or q ---> r or r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning H: Γ ⊢i r or r ---> r using BasicReasoning
?ψ =
foldr patt_imp
((p ---> r) ---> (q ---> r) ---> p or q ---> r or r)
[]
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H1: Γ
⊢i (p ---> r) --->
(q ---> r) ---> p or q ---> r or r
using BasicReasoning H: Γ ⊢i r or r ---> r using BasicReasoning
?ψ =
foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++ [p or q])
reflexivity.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp ?g'
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
well_formed (r or r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
well_formed ?g'
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Γ ⊢i r or r ---> ?g'using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp r
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
well_formed (r or r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
well_formed r
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp r
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
apply H1.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
well_formed (r or r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r H: Γ ⊢i r or r ---> r using BasicReasoning H1: Γ
⊢i foldr patt_imp (r or r)
((([] ++ [p ---> r]) ++ [q ---> r]) ++
[p or q]) using BasicReasoning
well_formed r
all: wf_auto2.Defined.
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i p ---> r using i
→ Γ ⊢i (q ---> r) ---> p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i p ---> r using i
→ Γ ⊢i (q ---> r) ---> p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
Γ ⊢i (q ---> r) ---> p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
Γ ⊢i ?ϕ₁ ---> (q ---> r) ---> p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
well_formed p
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
well_formed q
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i
well_formed r
all: wf_auto2.Defined.
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i p ---> r using i
→ Γ ⊢i q ---> r using i
→ Γ ⊢i p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i p ---> r using i
→ Γ ⊢i q ---> r using i
→ Γ ⊢i p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
Γ ⊢i p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
Γ ⊢i ?ϕ₁ ---> p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
Γ ⊢i (q ---> r) ---> p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
well_formed p
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
well_formed q
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
well_formed r
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
Γ ⊢i p ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
well_formed p
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
well_formed q
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i
well_formed r
all: wf_auto2.Defined.
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i p ---> r using i
→ Γ ⊢i q ---> r using i
→ Γ ⊢i p or q using i → Γ ⊢i r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i p ---> r using i
→ Γ ⊢i q ---> r using i
→ Γ ⊢i p or q using i → Γ ⊢i r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
Γ ⊢i r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
Γ ⊢i ?ϕ₁ ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
Γ ⊢i p or q ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
well_formed p
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
well_formed q
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
well_formed r
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
Γ ⊢i p ---> r using i
Σ: Signature Γ: Theory p, q, r: Pattern i: ProofInfo WFp: well_formed p WHq: well_formed q WFr: well_formed r H: Γ ⊢i p ---> r using i H0: Γ ⊢i q ---> r using i H1: Γ ⊢i p or q using i
Γ ⊢i q ---> r using i
all: assumption.Defined.
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo
Pattern.wf l
→ well_formed a
→ well_formed g
→ Γ ⊢i a using i
→ Γ
⊢i foldr patt_imp g (a :: l) --->
foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo
Pattern.wf l
→ well_formed a
→ well_formed g
→ Γ ⊢i a using i
→ Γ
⊢i foldr patt_imp g (a :: l) --->
foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo wfl: Pattern.wf l wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i
Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i
Γ ⊢i foldr patt_imp g [a] ---> foldr patt_imp g []
using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfl: Pattern.wf (a0 :: l) wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l → Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l using i
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i
Γ ⊢i foldr patt_imp g [a] ---> foldr patt_imp g []
using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i
Γ ⊢i (a ---> g) ---> g using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i H: Γ ⊢i a ---> (a ---> g) ---> g using BasicReasoning
Γ ⊢i (a ---> g) ---> g using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i H: Γ ⊢i a ---> (a ---> g) ---> g using BasicReasoning
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i H: Γ ⊢i a ---> (a ---> g) ---> g using BasicReasoning
Γ ⊢i ?ϕ₁ ---> (a ---> g) ---> g using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i H: Γ ⊢i a ---> (a ---> g) ---> g using BasicReasoning
Γ ⊢i a ---> (a ---> g) ---> g using i
Σ: Signature Γ: Theory a, g: Pattern i: ProofInfo wfl: Pattern.wf [] wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i H: Γ ⊢i a ---> (a ---> g) ---> g using BasicReasoning
Γ ⊢i a ---> (a ---> g) ---> g using BasicReasoning
apply H.
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfl: Pattern.wf (a0 :: l) wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l → Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l using i
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfl: Pattern.wf (a0 :: l) wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l → Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfl: foldr andb true (map well_formed (a0 :: l)) wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l → Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfl: well_formed a0 && foldr andb true (map well_formed l) wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l → Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfl: well_formed a0 = true ∧ foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l → Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp g (a :: l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i foldr patt_imp g (a :: l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i foldr patt_imp g (a :: a0 :: l) --->
foldr patt_imp g (a0 :: l) using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
(* < change a0 and a in the LHS > *)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
apply reorder; wf_auto2.
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
((a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l) = ?ϕ
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
Γ ⊢i ?ϕusing i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
((a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l) = ?ϕ
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
((a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l) = foldr patt_imp ?ϕ []
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀ (hh'g : Pattern) (i : ProofInfo),
Pattern.wf []
→ Pattern.wf []
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ ⊢i foldr patt_imp g ([] ++ [h])
using i
→ Γ
⊢i foldr patt_imp g ([] ++ [h'])
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀ (hh'g : Pattern) (i : ProofInfo),
Pattern.wf []
→ Pattern.wf []
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ ⊢i h ---> g using i
→ Γ ⊢i h' ---> g using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀ (hh'g : Pattern) (i : ProofInfo),
Pattern.wf []
→ Pattern.wf []
→ well_formed h
→ well_formed h'
→ well_formed g
→ Γ ⊢i h' ---> h using i
→ Γ ⊢i h ---> g using i
→ Γ ⊢i h' ---> g using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀ (g0 : Pattern) (i : ProofInfo),
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed g0
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
g0 using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
g0 using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Pattern.wf []
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Pattern.wf []
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
well_formed (a0 ---> a ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
well_formed (a ---> a0 ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
well_formed (a0 ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning
apply H.
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using BasicReasoning H0: ∀i : ProofInfo,
Pattern.wf []
→ Pattern.wf []
→ well_formed
(a0 ---> a ---> foldr patt_imp g l)
→ well_formed
(a ---> a0 ---> foldr patt_imp g l)
→ well_formed
(a0 ---> foldr patt_imp g l)
→ Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> a ---> foldr patt_imp g l
using i
→ Γ
⊢i (a0 --->
a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
→ Γ
⊢i (a --->
a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l
using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
(* </change a0 and a > *)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i ?ϕ₁ --->
(a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i ?ϕ₁ --->
(a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ
⊢i ?ϕ₁ --->
(a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using BasicReasoning
apply modus_ponens; wf_auto2.
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l)
Γ ⊢i a using i
apply Ha.
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
well_formed ?A
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
well_formed (a0 ---> a ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
well_formed (a0 ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ ⊢i (a0 ---> a ---> foldr patt_imp g l) ---> ?Ausing i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ ⊢i ?A ---> a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
well_formed (a ---> a0 ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
well_formed (a0 ---> a ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
well_formed (a0 ---> foldr patt_imp g l)
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a ---> a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a ---> a0 ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory a0: Pattern l: list Pattern a, g: Pattern i: ProofInfo wfa0: well_formed a0 = true wfl: foldr andb true (map well_formed l) = true wfa: well_formed a wfg: well_formed g Ha: Γ ⊢i a using i IHl: Γ
⊢i (a ---> foldr patt_imp g l) --->
foldr patt_imp g l using i wfa0l: Pattern.wf (a0 :: l) H: Γ
⊢i (a ---> a0 ---> foldr patt_imp g l) --->
a0 ---> foldr patt_imp g l using i
Γ
⊢i (a0 ---> a ---> foldr patt_imp g l) --->
a ---> a0 ---> foldr patt_imp g l
using BasicReasoning
apply reorder; wf_auto2.Defined.
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo
Pattern.wf l
→ well_formed a
→ well_formed g
→ Γ ⊢i a using i
→ Γ ⊢i foldr patt_imp g (a :: l) using i
→ Γ ⊢i foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo
Pattern.wf l
→ well_formed a
→ well_formed g
→ Γ ⊢i a using i
→ Γ ⊢i foldr patt_imp g (a :: l) using i
→ Γ ⊢i foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Γ ⊢i foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g l using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Γ ⊢i foldr patt_imp g (a :: l) ---> foldr patt_imp g l
using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Pattern.wf l
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
well_formed a
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
well_formed g
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Γ ⊢i a using i
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
Pattern.wf l
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
well_formed a
Σ: Signature Γ: Theory l: list Pattern a, g: Pattern i: ProofInfo WFl: Pattern.wf l WFa: well_formed a WFg: well_formed g H: Γ ⊢i a using i H0: Γ ⊢i foldr patt_imp g (a :: l) using i
well_formed g
all: wf_auto2.Defined.
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo
Γ ⊢i h using i
→ Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo
Γ ⊢i h using i
→ Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
→ Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed
(mlConclusion
(Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
(name ∶ h) :: l--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i
)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i
well_formed g
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp g (patterns_of l) using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (patterns_of l) using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed g
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i h using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (h :: patterns_of l) using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed g
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i h using i
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed g
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (h :: patterns_of l)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
well_formed h
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (h :: patterns_of l)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (h :: patterns_of l)
Σ: Signature Γ: Theory l: list named_hypothesis name: string g, h: Pattern i: ProofInfo H: Γ ⊢i h using i H0: well_formed g
→ Pattern.wf (h :: patterns_of l)
→ Γ ⊢i h ---> foldr patt_imp g (patterns_of l)
using i wfg: well_formed g wfl: Pattern.wf (patterns_of l)
Pattern.wf (h :: patterns_of l)
abstract (
unfold Pattern.wf;
simpl;
pose (tmp := proj1_sig H);
apply proved_impl_wf in tmp;
rewrite tmp;
simpl;
exact wfl
).}Defined.Tactic Notation"mlAdd"constr(n) "as"constr(name') :=
_failIfUsed name';
match goal with
| |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm?Ctx?l?g?i) =>
apply (@MLGoal_add Sgm Ctx l name' g _ i n)
end.Tactic Notation"mlAdd"constr(n) :=
lethyps := _getHypNames inletname := evalcbvin (fresh hyps) in
mlAdd n as name.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ ⊢i h ---> g using i
→ Γ ⊢i h using i → Γ ⊢i g using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed h
→ Γ ⊢i h ---> g using i
→ Γ ⊢i h using i → Γ ⊢i g using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
Γ ⊢i g using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
well_formed g
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
Γ Ⱶ
g
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
well_formed g
wf_auto2.
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
Γ Ⱶ
g
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
Γ Ⱶ
"H0" ∶ h,
--------------------------------------
g
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
Γ Ⱶ
"0" ∶ h ---> g,
"H0" ∶ h,
--------------------------------------
g
using i
Σ: Signature Γ: Theory l: list Pattern g, h: Pattern i: ProofInfo WFl: Pattern.wf l WFg: well_formed g WFh: well_formed h H: Γ ⊢i h ---> g using i H0: Γ ⊢i h using i
Γ Ⱶ
"0" ∶ h ---> g,
"H0" ∶ h,
--------------------------------------
h
using i
mlExact "H0".Defined.
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf l1 wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h
Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory l2: list Pattern g, h: Pattern wfl1: Pattern.wf [] wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h
Γ
⊢i foldr patt_imp g l2 ---> h ---> foldr patt_imp g l2
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf (a :: l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory l2: list Pattern g, h: Pattern wfl1: Pattern.wf [] wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h
Γ
⊢i foldr patt_imp g l2 ---> h ---> foldr patt_imp g l2
using BasicReasoning
apply P1; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: Pattern.wf (a :: l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: foldr andb true (map well_formed (a :: l1)) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: well_formed a &&
foldr andb true (map well_formed l1) wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfl1: well_formed a = true
∧ foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Pattern.wf l1
→ Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning H1: Γ
⊢i a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
well_formed
(a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ h :: l2))
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ Ⱶ
a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
well_formed
(a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ h :: l2))
wf_auto2.
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ Ⱶ
a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ Ⱶ
"H0" ∶ foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2),
--------------------------------------
a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ Ⱶ
"H0" ∶ foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2),
"H1" ∶ a,
--------------------------------------
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
mlExact "H0".
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning H1: Γ
⊢i a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i (a ---> foldr patt_imp g (l1 ++ l2)) --->
a ---> foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l1, l2: list Pattern g, h: Pattern wfa: well_formed a = true wfl1: foldr andb true (map well_formed l1) = true wfl2: Pattern.wf l2 wfg: well_formed g wfh: well_formed h IHl1: Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning H1: Γ
⊢i a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ h :: l2)
using BasicReasoning
apply H1.Defined.
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
→ Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2)
using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo
Pattern.wf l1
→ Pattern.wf l2
→ well_formed g
→ well_formed h
→ Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
→ Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2)
using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo H: Pattern.wf l1 H0: Pattern.wf l2 H1: well_formed g H2: well_formed h H3: Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Γ ⊢i foldr patt_imp g (l1 ++ [h] ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo H: Pattern.wf l1 H0: Pattern.wf l2 H1: well_formed g H2: well_formed h H3: Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo H: Pattern.wf l1 H0: Pattern.wf l2 H1: well_formed g H2: well_formed h H3: Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l1 ++ [h] ++ l2)
using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo H: Pattern.wf l1 H0: Pattern.wf l2 H1: well_formed g H2: well_formed h H3: Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2) using i
Σ: Signature Γ: Theory l1, l2: list Pattern g, h: Pattern i: ProofInfo H: Pattern.wf l1 H0: Pattern.wf l2 H1: well_formed g H2: well_formed h H3: Γ ⊢i foldr patt_imp g (l1 ++ l2) using i
Γ
⊢i foldr patt_imp g (l1 ++ l2) --->
foldr patt_imp g (l1 ++ [h] ++ l2)
using BasicReasoning
apply prf_clear_hyp; wf_auto2.Defined.
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo
Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
→ Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo
Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
→ Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed
(mlConclusion
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l1 ++ l2--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l1 ++ h :: l2--------------------------------------
g
using i
)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (patterns_of (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (patterns_of (l1 ++ l2))
using i
well_formed g
→ Pattern.wf (patterns_of (l1 ++ h :: l2))
→ Γ
⊢i foldr patt_imp g (patterns_of (l1 ++ h :: l2))
using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (patterns_of (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (patterns_of (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf (patterns_of (l1 ++ h :: l2))
Γ ⊢i foldr patt_imp g (patterns_of (l1 ++ h :: l2))
using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf (map nh_patt (l1 ++ h :: l2))
Γ ⊢i foldr patt_imp g (map nh_patt (l1 ++ h :: l2))
using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf (map nh_patt (l1 ++ h :: l2))
Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt (h :: l2)) using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt (h :: l2)) using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed g
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed (nh_patt h)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++
(fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2) using i
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed g
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed (nh_patt h)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
well_formed g
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed (nh_patt h)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf (map nh_patt l1)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf (map nh_patt l1)
Pattern.wf (map nh_patt l1)
exact wfl1hl2.
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed (nh_patt h)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf (map nh_patt l2)
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l2)
exact wfl1hl2.
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed (nh_patt h)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2)
well_formed (nh_patt h)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo H1: well_formed g
→ Pattern.wf (map nh_patt (l1 ++ l2))
→ Γ
⊢i foldr patt_imp g (map nh_patt (l1 ++ l2))
using i wfg: well_formed g wfl1hl2: well_formed (nh_patt h)
well_formed (nh_patt h)
exact wfl1hl2.
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++
nh_patt h :: map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
Σ: Signature Γ: Theory l1, l2: list named_hypothesis g: Pattern h: named_hypothesis i: ProofInfo wfg: well_formed g wfl1hl2: Pattern.wf
(map nh_patt l1 ++ map nh_patt l2) H1: well_formed g
→ Pattern.wf (map nh_patt l1 ++ map nh_patt l2)
→ Γ
⊢i foldr patt_imp g
(map nh_patt l1 ++ map nh_patt l2)
using i
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
foldr patt_imp r [p; q; p ---> q ---> r] = ?ϕ
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
foldr patt_imp r ([p] ++ [q; p ---> q ---> r] ++ []) =
?ϕ
reflexivity.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ
⊢i foldr patt_imp r
([p] ++ [q; p ---> q ---> r] ++ [])
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ
⊢i foldr patt_imp r
([p] ++ [p ---> q ---> r; q] ++ [])
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ ⊢i p ---> (p ---> q ---> r) ---> q ---> r
using BasicReasoning
apply modus_ponens; wf_auto2.Defined.
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern
Pattern.wf l
→ well_formed g
→ well_formed x
→ Γ
⊢i foldr patt_imp g (x :: l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern
Pattern.wf l
→ well_formed g
→ well_formed x
→ Γ
⊢i foldr patt_imp g (x :: l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern wfl: Pattern.wf l wfg: well_formed g wfx: well_formed x
Γ
⊢i foldr patt_imp g (x :: l) --->
foldr patt_imp g (l ++ [x]) using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfl: Pattern.wf [] wfg: well_formed g wfx: well_formed x
Γ
⊢i foldr patt_imp g [x] --->
foldr patt_imp g ([] ++ [x]) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfl: Pattern.wf [] wfg: well_formed g wfx: well_formed x
Γ
⊢i foldr patt_imp g [x] --->
foldr patt_imp g ([] ++ [x]) using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfl: Pattern.wf [] wfg: well_formed g wfx: well_formed x
Γ ⊢i (x ---> g) ---> x ---> g using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfl: Pattern.wf [] wfg: well_formed g wfx: well_formed x
well_formed (x ---> g)
wf_auto2.
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfl: Pattern.wf (a :: l) wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfl: foldr andb true (map well_formed (a :: l)) wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfl: well_formed a && foldr andb true (map well_formed l) wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfl: well_formed a = true ∧ foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp g (x :: a :: l) --->
foldr patt_imp g ((a :: l) ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i foldr patt_imp g (x :: l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (x ---> a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g (l ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (x ---> a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g (l ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
((x ---> a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g (l ++ [x])) = ?ϕ
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
((x ---> a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g (l ++ [x])) = ?ϕ
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
foldr patt_imp
((x ---> a ---> foldr patt_imp g l) --->
a ---> foldr patt_imp g (l ++ [x])) [] = ?ϕ
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
foldr patt_imp (foldr patt_imp g (l ++ [x]))
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++ [a]) =
?ϕ
reflexivity.
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp (foldr patt_imp g (l ++ [x]))
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++
[a]) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Pattern.wf
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++ [a])
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed ?g
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed (foldr patt_imp g (l ++ [x]))
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ ⊢i ?g ---> foldr patt_imp g (l ++ [x])
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp ?g
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++
[a]) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ ⊢i ?g ---> foldr patt_imp g (l ++ [x])
using BasicReasoning
apply IHl.
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Pattern.wf
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++ [a])
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed (x ---> foldr patt_imp g l)
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed (foldr patt_imp g (l ++ [x]))
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp (x ---> foldr patt_imp g l)
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++
[a]) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp (x ---> foldr patt_imp g l)
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++
[a]) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
foldr patt_imp (x ---> foldr patt_imp g l)
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++ [a]) =
?ϕ
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ
⊢i (x ---> foldr patt_imp g l) --->
foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
foldr patt_imp (x ---> foldr patt_imp g l)
(([] ++ [x ---> a ---> foldr patt_imp g l]) ++ [a]) =
?ϕ
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
foldr patt_imp (foldr patt_imp g l)
((([] ++ [x ---> a ---> foldr patt_imp g l]) ++ [a]) ++
[x]) = ?ϕ
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
foldr patt_imp (foldr patt_imp g l)
([x ---> a ---> foldr patt_imp g l] ++ [a; x] ++ []) =
?ϕ
reflexivity.
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g wfx: well_formed x IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp (foldr patt_imp g l)
([x ---> a ---> foldr patt_imp g l] ++
[a; x] ++ []) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g = true wfx: well_formed x = true IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l) = true
Γ
⊢i foldr patt_imp (foldr patt_imp g l)
([x ---> a ---> foldr patt_imp g l] ++
[x; a] ++ []) using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g = true wfx: well_formed x = true IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l) = true
Γ
⊢i (x ---> a ---> foldr patt_imp g l) --->
x ---> a ---> foldr patt_imp g l
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern l: list Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfg: well_formed g = true wfx: well_formed x = true IHl: Γ ⊢i (x ---> foldr patt_imp g l) ---> foldr patt_imp g (l ++ [x])
using BasicReasoning wfal: Pattern.wf (a :: l) = true
well_formed (x ---> a ---> foldr patt_imp g l)
wf_auto2.Defined.
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed x
→ Γ ⊢i foldr patt_imp g (x :: l) using i
→ Γ ⊢i foldr patt_imp g (l ++ [x]) using i
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo
Pattern.wf l
→ well_formed g
→ well_formed x
→ Γ ⊢i foldr patt_imp g (x :: l) using i
→ Γ ⊢i foldr patt_imp g (l ++ [x]) using i
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo WFl: Pattern.wf l WFG: well_formed g WFx: well_formed x H: Γ ⊢i foldr patt_imp g (x :: l) using i
Γ ⊢i foldr patt_imp g (l ++ [x]) using i
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo WFl: Pattern.wf l WFG: well_formed g WFx: well_formed x H: Γ ⊢i foldr patt_imp g (x :: l) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo WFl: Pattern.wf l WFG: well_formed g WFx: well_formed x H: Γ ⊢i foldr patt_imp g (x :: l) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (l ++ [x]) using i
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo WFl: Pattern.wf l WFG: well_formed g WFx: well_formed x H: Γ ⊢i foldr patt_imp g (x :: l) using i
Γ
⊢i foldr patt_imp g (x :: l) --->
foldr patt_imp g (l ++ [x]) using i
Σ: Signature Γ: Theory g, x: Pattern l: list Pattern i: ProofInfo WFl: Pattern.wf l WFG: well_formed g WFx: well_formed x H: Γ ⊢i foldr patt_imp g (x :: l) using i
Γ
⊢i foldr patt_imp g (x :: l) --->
foldr patt_imp g (l ++ [x]) using BasicReasoning
apply reorder_last_to_head; wf_auto2.Defined.(* Iterated modus ponens. For l = [x₁, ..., xₙ], it says that Γ ⊢i ((x₁ -> ... -> xₙ -> (x₁ -> ... -> xₙ -> r)) -> r)*)
Σ: Signature Γ: Theory l: list Pattern r: Pattern
Pattern.wf l
→ well_formed r
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern r: Pattern
Pattern.wf l
→ well_formed r
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern r: Pattern wfl: Pattern.wf l wfr: well_formed r
Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory r: Pattern wfl: Pattern.wf [] wfr: well_formed r
Γ ⊢i foldr patt_imp r ([] ++ [foldr patt_imp r []])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfl: Pattern.wf (a :: l) wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory r: Pattern wfl: Pattern.wf [] wfr: well_formed r
Γ ⊢i foldr patt_imp r ([] ++ [foldr patt_imp r []])
using BasicReasoning
Σ: Signature Γ: Theory r: Pattern wfl: Pattern.wf [] wfr: well_formed r
Γ ⊢i r ---> r using BasicReasoning
Σ: Signature Γ: Theory r: Pattern wfl: Pattern.wf [] wfr: well_formed r
well_formed r
exact wfr.
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfl: Pattern.wf (a :: l) wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfl: Pattern.wf (a :: l) wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfl: foldr andb true (map well_formed (a :: l)) wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfl: well_formed a && foldr andb true (map well_formed l) wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfl: well_formed a = true ∧ foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Pattern.wf l
→ Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r
((a :: l) ++ [foldr patt_imp r (a :: l)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
(a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [foldr patt_imp r l])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
(a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
(a --->
foldr patt_imp
(foldr patt_imp r [a ---> foldr patt_imp r l]) l) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
(a --->
foldr patt_imp ((a ---> foldr patt_imp r l) ---> r) l) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
(a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
(a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])) =
?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ ⊢i ?ψusing BasicReasoning
Γ
⊢i a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [foldr patt_imp r l])
using BasicReasoning wfal: Pattern.wf (a :: l)
?ψ = foldr patt_imp r (l ++ [foldr patt_imp r l])
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
?ψ = foldr patt_imp r (l ++ [foldr patt_imp r l])
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r IHl: Γ ⊢i foldr patt_imp r (l ++ [foldr patt_imp r l]) using BasicReasoning wfal: Pattern.wf (a :: l)
?ψ =
foldr patt_imp (foldr patt_imp r [foldr patt_imp r l])
l
reflexivity.
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp
(foldr patt_imp r [foldr patt_imp r l]) l
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
well_formed a
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
well_formed ?B
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
well_formed
(foldr patt_imp r (l ++ [a ---> foldr patt_imp r l]))
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ
⊢i ?B --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ ⊢i a ---> ?Busing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ
⊢i ?B --->
foldr patt_imp r (l ++ [a ---> foldr patt_imp r l])
using BasicReasoning
apply reorder_last_to_head; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
well_formed a
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
well_formed
(foldr patt_imp r ((a ---> foldr patt_imp r l) :: l))
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
well_formed
(foldr patt_imp r (l ++ [a ---> foldr patt_imp r l]))
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp r ((a ---> foldr patt_imp r l) :: l)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ
⊢i a --->
foldr patt_imp r ((a ---> foldr patt_imp r l) :: l)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfr: well_formed r wfal: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (foldr patt_imp r l ---> r) l
using BasicReasoning
Γ
⊢i a --->
(a ---> foldr patt_imp r l) ---> foldr patt_imp r l
using BasicReasoning
apply modus_ponens; wf_auto2.Defined.
Σ: Signature Γ: Theory p, q, r: Pattern
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i (p and q ---> r) ---> p ---> q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i (p and q ---> r) ---> p ---> q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ ⊢i (p and q ---> r) ---> p ---> q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed ((p and q ---> r) ---> p ---> q ---> r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
(p and q ---> r) ---> p ---> q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed ((p and q ---> r) ---> p ---> q ---> r)
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
(p and q ---> r) ---> p ---> q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ p and q ---> r,
--------------------------------------
p ---> q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ p and q ---> r,
"H2" ∶ p,
--------------------------------------
q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ p and q ---> r,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
! (! p or ! q)
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! p or ! q,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed (! ! p)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
--------------------------------------
! ! p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed (! ! p)
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
--------------------------------------
! ! p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
--------------------------------------
! ! p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H6" ∶ p ---> ! ! p,
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
--------------------------------------
! ! p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H6" ∶ p ---> ! ! p,
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
--------------------------------------
p
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
well_formed (! q)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
"H6" ∶ ! q,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p)
well_formed (! q)
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
"H6" ∶ ! q,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
! q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
--------------------------------------
! ! p
using BasicReasoning
mlExact "H5".
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
"H6" ∶ ! q,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed (! ! p) Hwf0: well_formed (! q)
Γ Ⱶ
"H0" ∶ ! (! p or ! q) ---> r,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ ! ! p ---> ! q,
"H5" ∶ ! ! p,
"H6" ∶ q ---> ⊥,
--------------------------------------
q
using BasicReasoning
mlExact "H3".Defined.
Σ: Signature Γ: Theory p, q, r: Pattern
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i (p ---> q ---> r) ---> p and q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern
well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i (p ---> q ---> r) ---> p and q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ ⊢i (p ---> q ---> r) ---> p and q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed ((p ---> q ---> r) ---> p and q ---> r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
(p ---> q ---> r) ---> p and q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed ((p ---> q ---> r) ---> p and q ---> r)
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
(p ---> q ---> r) ---> p and q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
--------------------------------------
p and q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed p
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r
well_formed p
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H2" ∶ p and q ---> p,
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H2" ∶ p and q ---> p,
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
--------------------------------------
p and q
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
well_formed q
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p
well_formed q
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H4" ∶ p and q ---> q,
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H4" ∶ p and q ---> q,
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
--------------------------------------
p and q
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
r
using BasicReasoning
(* This pattern is basically an "apply ... in" *)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
well_formed (q ---> r)
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ q ---> r,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q
well_formed (q ---> r)
wf_auto2.
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ q ---> r,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
q ---> r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
--------------------------------------
p
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ q ---> r,
--------------------------------------
r
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfp: well_formed p wfq: well_formed q wfr: well_formed r Hwf: well_formed p Hwf0: well_formed q Hwf1: well_formed (q ---> r)
Γ Ⱶ
"H0" ∶ p ---> q ---> r,
"H1" ∶ p and q,
"H2" ∶ p,
"H3" ∶ q,
"H4" ∶ q ---> r,
--------------------------------------
q
using BasicReasoning
mlExact "H3".Defined.
Σ: Signature Γ: Theory l: list Pattern p, q, r: Pattern
Pattern.wf l
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern p, q, r: Pattern
Pattern.wf l
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory l: list Pattern p, q, r: Pattern wfl: Pattern.wf l wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfl: Pattern.wf [] wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ
⊢i foldr patt_imp r ([] ++ [p]) --->
foldr patt_imp r ([] ++ [q]) --->
foldr patt_imp r ([] ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfl: Pattern.wf (a :: l) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfl: Pattern.wf [] wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ
⊢i foldr patt_imp r ([] ++ [p]) --->
foldr patt_imp r ([] ++ [q]) --->
foldr patt_imp r ([] ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfl: Pattern.wf [] wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ ⊢i (p ---> r) ---> (q ---> r) ---> p or q ---> r
using BasicReasoning
apply prf_disj_elim; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfl: Pattern.wf (a :: l) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfl: Pattern.wf (a :: l) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfl: foldr andb true (map well_formed (a :: l)) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfl: well_formed a && foldr andb true (map well_formed l) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfl: well_formed a = true ∧ foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i foldr patt_imp r ((a :: l) ++ [p]) --->
foldr patt_imp r ((a :: l) ++ [q]) --->
foldr patt_imp r ((a :: l) ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp r (l ++ [p])) --->
(a ---> foldr patt_imp r (l ++ [q])) --->
a ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed
((a ---> foldr patt_imp r (l ++ [p])) --->
(a ---> foldr patt_imp r (l ++ [q])) --->
a ---> foldr patt_imp r (l ++ [p or q]))
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ Ⱶ
(a ---> foldr patt_imp r (l ++ [p])) --->
(a ---> foldr patt_imp r (l ++ [q])) --->
a ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed
((a ---> foldr patt_imp r (l ++ [p])) --->
(a ---> foldr patt_imp r (l ++ [q])) --->
a ---> foldr patt_imp r (l ++ [p or q]))
wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ Ⱶ
(a ---> foldr patt_imp r (l ++ [p])) --->
(a ---> foldr patt_imp r (l ++ [q])) --->
a ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
--------------------------------------
(a ---> foldr patt_imp r (l ++ [q])) --->
a ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
--------------------------------------
a ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed (foldr patt_imp r (l ++ [p]))
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
--------------------------------------
foldr patt_imp r (l ++ [p])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l)
well_formed (foldr patt_imp r (l ++ [p]))
wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
--------------------------------------
foldr patt_imp r (l ++ [p])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
--------------------------------------
foldr patt_imp r (l ++ [p])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
well_formed (foldr patt_imp r (l ++ [q]))
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
foldr patt_imp r (l ++ [q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p]))
well_formed (foldr patt_imp r (l ++ [q]))
wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
foldr patt_imp r (l ++ [q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
foldr patt_imp r (l ++ [q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
--------------------------------------
a
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
"H6" ∶ foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q]))
well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
wf_auto2.
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
"H6" ∶ foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
--------------------------------------
foldr patt_imp r (l ++ [p])
using BasicReasoning
mlExact "H4".
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
"H6" ∶ foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
--------------------------------------
foldr patt_imp r (l ++ [p or q])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl: foldr andb true (map well_formed l) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl: Γ
⊢i foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) ---> foldr patt_imp r (l ++ [p or q])
using BasicReasoning wfal: Pattern.wf (a :: l) Hwf: well_formed (foldr patt_imp r (l ++ [p])) Hwf0: well_formed (foldr patt_imp r (l ++ [q])) Hwf1: well_formed
(foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]))
Γ Ⱶ
"H3" ∶ foldr patt_imp r (l ++ [p]) --->
foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
"H0" ∶ a ---> foldr patt_imp r (l ++ [p]),
"H1" ∶ a ---> foldr patt_imp r (l ++ [q]),
"H2" ∶ a,
"H4" ∶ foldr patt_imp r (l ++ [p]),
"H5" ∶ foldr patt_imp r (l ++ [q]),
"H6" ∶ foldr patt_imp r (l ++ [q]) --->
foldr patt_imp r (l ++ [p or q]),
--------------------------------------
foldr patt_imp r (l ++ [q])
using BasicReasoning
mlExact "H5".Defined.
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern wfl₁: Pattern.wf l₁ wfl₂: Pattern.wf l₂ wfp: well_formed p wfq: well_formed q wfr: well_formed r
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory l₂: list Pattern p, q, r: Pattern wfl₂: Pattern.wf l₂ wfp: well_formed p wfq: well_formed q wfr: well_formed r
∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfl₂: Pattern.wf [] wfp: well_formed p wfq: well_formed q wfr: well_formed r l₁: list Pattern wfl₁: Pattern.wf l₁
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ []) --->
foldr patt_imp r (l₁ ++ [q] ++ []) --->
foldr patt_imp r (l₁ ++ [p or q] ++ [])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfl₂: Pattern.wf (a :: l₂) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfl₂: Pattern.wf [] wfp: well_formed p wfq: well_formed q wfr: well_formed r l₁: list Pattern wfl₁: Pattern.wf l₁
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ []) --->
foldr patt_imp r (l₁ ++ [q] ++ []) --->
foldr patt_imp r (l₁ ++ [p or q] ++ [])
using BasicReasoning
Σ: Signature Γ: Theory p, q, r: Pattern wfl₂: Pattern.wf [] wfp: well_formed p wfq: well_formed q wfr: well_formed r l₁: list Pattern wfl₁: Pattern.wf l₁
Γ
⊢i foldr patt_imp r (l₁ ++ [p]) --->
foldr patt_imp r (l₁ ++ [q]) --->
foldr patt_imp r (l₁ ++ [p or q])
using BasicReasoning
apply prf_disj_elim_iter; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfl₂: Pattern.wf (a :: l₂) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfl₂: Pattern.wf (a :: l₂) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfl₂: foldr andb true (map well_formed (a :: l₂)) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfl₂: well_formed a &&
foldr andb true (map well_formed l₂) wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfl₂: well_formed a = true
∧ foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ a :: l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
using BasicReasoning
(* Or we can swap p and a (move a to the end of l_1) *)
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(A --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(A --->
B ---> foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(A ---> B ---> C) = ?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i A ---> B ---> C using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(A ---> B ---> C) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(A ---> B ---> C) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp (A ---> B ---> C) [] = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp (B ---> C) ([] ++ [A]) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp C (([] ++ [A]) ++ [B]) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp C [A; B] = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp C
[foldr patt_imp r (l₁ ++ p :: a :: l₂); B] = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp C
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)] = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂))
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)] = ?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂))
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)]
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)]
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed ?g
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i ?g --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp ?g
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)]
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i ?g --->
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(?g ---> foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(?g ---> foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)) =
?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(?g ---> foldr patt_imp r (l₁ ++ [p or q; a] ++ l₂)) =
?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i ?g ---> foldr patt_imp r (l₁ ++ [p or q; a] ++ l₂)
using BasicReasoning
apply prf_reorder_iter; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)]
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ [a; p or q] ++ l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ [a; p or q] ++ l₂))
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)]
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ [a; p or q] ++ l₂))
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)]
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) [] =
?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
(([] ++ [foldr patt_imp r (l₁ ++ p :: a :: l₂)]) ++
[foldr patt_imp r (l₁ ++ q :: a :: l₂)]) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ q :: a :: l₂)] = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ q :: a :: l₂)]) = ?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ q :: a :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf [foldr patt_imp r (l₁ ++ p :: a :: l₂)]
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf []
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ a :: q :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ q :: a :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: q :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ q :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: q :: l₂)
using BasicReasoning
apply prf_reorder_iter; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf [foldr patt_imp r (l₁ ++ p :: a :: l₂)]
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf []
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ a :: q :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ q :: a :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ a :: q :: l₂)]) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([foldr patt_imp r (l₁ ++ p :: a :: l₂)] ++
[foldr patt_imp r (l₁ ++ a :: q :: l₂)]) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([] ++
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ a :: q :: l₂)]) = ?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([] ++
[foldr patt_imp r (l₁ ++ p :: a :: l₂);
foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf []
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf [foldr patt_imp r (l₁ ++ a :: q :: l₂)]
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ a :: p :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ p :: a :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: p :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([] ++
[foldr patt_imp r (l₁ ++ a :: p :: l₂);
foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ p :: a :: l₂) --->
foldr patt_imp r (l₁ ++ a :: p :: l₂)
using BasicReasoning
apply prf_reorder_iter; wf_auto2.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf []
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Pattern.wf [foldr patt_imp r (l₁ ++ a :: q :: l₂)]
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ a :: p :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed (foldr patt_imp r (l₁ ++ p :: a :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
well_formed
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([] ++
[foldr patt_imp r (l₁ ++ a :: p :: l₂);
foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp
(foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂))
([] ++
[foldr patt_imp r (l₁ ++ a :: p :: l₂);
foldr patt_imp r (l₁ ++ a :: q :: l₂)])
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r (l₁ ++ a :: p :: l₂) --->
foldr patt_imp r (l₁ ++ a :: q :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r (l₁ ++ a :: p :: l₂) --->
foldr patt_imp r (l₁ ++ a :: q :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ ⊢i ?ϕusing BasicReasoning
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r (l₁ ++ a :: p :: l₂) --->
foldr patt_imp r (l₁ ++ a :: q :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r ((l₁ ++ [a]) ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ a :: q :: l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r ((l₁ ++ [a]) ++ [p] ++ l₂) --->
foldr patt_imp r ((l₁ ++ [a]) ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ a :: (p or q) :: l₂)) = ?ϕ
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
(foldr patt_imp r ((l₁ ++ [a]) ++ [p] ++ l₂) --->
foldr patt_imp r ((l₁ ++ [a]) ++ [q] ++ l₂) --->
foldr patt_imp r ((l₁ ++ [a]) ++ [p or q] ++ l₂)) =
?ϕ
reflexivity.
Σ: Signature Γ: Theory a: Pattern l₂: list Pattern p, q, r: Pattern wfa: well_formed a = true wfl₂: foldr andb true (map well_formed l₂) = true wfp: well_formed p wfq: well_formed q wfr: well_formed r IHl₂: Pattern.wf l₂
→ ∀l₁ : list Pattern,
Pattern.wf l₁
→ Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r
(l₁ ++ [p or q] ++ l₂)
using BasicReasoning l₁: list Pattern wfl₁: Pattern.wf l₁ wfal₂: Pattern.wf (a :: l₂) A: Pattern HeqA: A = foldr patt_imp r (l₁ ++ p :: a :: l₂) B: Pattern HeqB: B = foldr patt_imp r (l₁ ++ q :: a :: l₂) C: Pattern HeqC: C =
foldr patt_imp r (l₁ ++ (p or q) :: a :: l₂)
Γ
⊢i foldr patt_imp r ((l₁ ++ [a]) ++ [p] ++ l₂) --->
foldr patt_imp r ((l₁ ++ [a]) ++ [q] ++ l₂) --->
foldr patt_imp r ((l₁ ++ [a]) ++ [p or q] ++ l₂)
using BasicReasoning
apply IHl₂; wf_auto2.Defined.
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂)
using i
→ Γ
⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂)
using i
→ Γ
⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i
Γ
⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i
Γ
⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) --->
foldr patt_imp r (l₁ ++ [q] ++ l₂) --->
foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using BasicReasoning
apply prf_disj_elim_iter_2; wf_auto2.Defined.
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂)
using i
→ Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂)
using i
→ Γ
⊢i foldr patt_imp r
(l₁ ++ [p or q] ++ l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo
Pattern.wf l₁
→ Pattern.wf l₂
→ well_formed p
→ well_formed q
→ well_formed r
→ Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂)
using i
→ Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂)
using i
→ Γ
⊢i foldr patt_imp r
(l₁ ++ [p or q] ++ l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i H0: Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) using i
Γ ⊢i foldr patt_imp r (l₁ ++ [p or q] ++ l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i H0: Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i H0: Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i H0: Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp r (l₁ ++ [p or q] ++ l₂)
using i
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i H0: Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) using i
Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i
apply H.
Σ: Signature Γ: Theory l₁, l₂: list Pattern p, q, r: Pattern i: ProofInfo WFl1: Pattern.wf l₁ WFl2: Pattern.wf l₂ WFp: well_formed p WFq: well_formed q WFr: well_formed r H: Γ ⊢i foldr patt_imp r (l₁ ++ [p] ++ l₂) using i H0: Γ ⊢i foldr patt_imp r (l₁ ++ [q] ++ l₂) using i
Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
→ Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
→ Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
→ Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
→ Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H1: Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
H2: Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H1: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (pn ∶ p,
) ++ l₂--------------------------------------
r
using i
) H2: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (qn ∶ q,
) ++ l₂--------------------------------------
r
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (pqn ∶ p or q,
) ++ l₂--------------------------------------
r
using i
)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H1: well_formed r
→ Pattern.wf (patterns_of (l₁ ++ (pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(patterns_of (l₁ ++ (pn ∶ p) :: l₂))
using i H2: well_formed r
→ Pattern.wf (patterns_of (l₁ ++ (qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(patterns_of (l₁ ++ (qn ∶ q) :: l₂))
using i
well_formed r
→ Pattern.wf
(patterns_of (l₁ ++ (pqn ∶ p or q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(patterns_of (l₁ ++ (pqn ∶ p or q) :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H1: well_formed r
→ Pattern.wf (patterns_of (l₁ ++ (pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(patterns_of (l₁ ++ (pn ∶ p) :: l₂))
using i H2: well_formed r
→ Pattern.wf (patterns_of (l₁ ++ (qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(patterns_of (l₁ ++ (qn ∶ q) :: l₂))
using i wfr: well_formed r Hwf: Pattern.wf (patterns_of (l₁ ++ (pqn ∶ p or q) :: l₂))
Γ
⊢i foldr patt_imp r
(patterns_of (l₁ ++ (pqn ∶ p or q) :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H1: well_formed r
→ Pattern.wf (map nh_patt (l₁ ++ (pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt (l₁ ++ (pn ∶ p) :: l₂))
using i H2: well_formed r
→ Pattern.wf (map nh_patt (l₁ ++ (qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt (l₁ ++ (qn ∶ q) :: l₂))
using i wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂))
Γ
⊢i foldr patt_imp r
(map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂))
using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H1: well_formed r
→ Pattern.wf (map nh_patt (l₁ ++ (pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt (l₁ ++ (pn ∶ p) :: l₂))
using i H2: well_formed r
→ Pattern.wf (map nh_patt (l₁ ++ (qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt (l₁ ++ (qn ∶ q) :: l₂))
using i wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂))
Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pqn ∶ p or q) :: l₂)) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo H2: well_formed r
→ Pattern.wf (map nh_patt (l₁ ++ (qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt (l₁ ++ (qn ∶ q) :: l₂))
using i wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i
Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pqn ∶ p or q) :: l₂)) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pqn ∶ p or q) :: l₂)) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Pattern.wf (map nh_patt l₁)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed p
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed q
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed r
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
[p] ++
(fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
[q] ++
(fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Pattern.wf (map nh_patt l₁)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed p
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed q
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed r
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
[p] ++
(fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
well_formed r
Σ: Signature Γ: Theory l₁, l₂: list named_hypothesis pn: string p: Pattern qn: string q: Pattern pqn: string r: Pattern i: ProofInfo wfr: well_formed r Hwf: Pattern.wf (map nh_patt (l₁ ++ (pqn ∶ p or q) :: l₂)) H1: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((pn ∶ p) :: l₂)) using i H2: well_formed r
→ Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂))
→ Γ
⊢i foldr patt_imp r
(map nh_patt l₁ ++
map nh_patt ((qn ∶ q) :: l₂)) using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed p
→ well_formed q
→ well_formed c
→ Γ ⊢i a ---> p ---> b ---> c using i
→ Γ ⊢i a ---> q ---> b ---> c using i
→ Γ ⊢i a ---> p or q ---> b ---> c using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed p
→ well_formed q
→ well_formed c
→ Γ ⊢i a ---> p ---> b ---> c using i
→ Γ ⊢i a ---> q ---> b ---> c using i
→ Γ ⊢i a ---> p or q ---> b ---> c using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ ⊢i a ---> p or q ---> b ---> c using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
well_formed (a ---> p or q ---> b ---> c)
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
a ---> p or q ---> b ---> c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
well_formed (a ---> p or q ---> b ---> c)
wf_auto2.
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
a ---> p or q ---> b ---> c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
--------------------------------------
p or q ---> b ---> c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p or q,
--------------------------------------
b ---> c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ p or q,
"H2" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"0" ∶ p,
"H2" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"1" ∶ q,
"H2" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"0" ∶ p,
"H2" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ ⊢i a ---> p ---> b ---> c using i
apply H.
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"1" ∶ q,
"H2" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, p, q, c: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b WFp: well_formed p WFq: well_formed q WFc: well_formed c H: Γ ⊢i a ---> p ---> b ---> c using i H0: Γ ⊢i a ---> q ---> b ---> c using i
Γ ⊢i a ---> q ---> b ---> c using i
apply H0.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B using i
→ Γ ⊢i B ---> A using i → Γ ⊢i A <---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B using i
→ Γ ⊢i B ---> A using i → Γ ⊢i A <---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimplB: Γ ⊢i A ---> B using i BimplA: Γ ⊢i B ---> A using i
Γ ⊢i A <---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B AimplB: Γ ⊢i A ---> B using i BimplA: Γ ⊢i B ---> A using i
well_formed A
→ well_formed B
→ Γ ⊢i A <---> B using i → Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A <---> B using i → Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A <---> B using i
Γ ⊢i A ---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i (A ---> B) and (B ---> A) using i
Γ ⊢i A ---> B using i
apply pf_conj_elim_l_meta in H; try_wfauto2; assumption.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A <---> B using i → Γ ⊢i B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A <---> B using i → Γ ⊢i B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A <---> B using i
Γ ⊢i B ---> A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i (A ---> B) and (B ---> A) using i
Γ ⊢i B ---> A using i
apply pf_conj_elim_r_meta in H; try_wfauto2; assumption.Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ (Γ ⊢i A <---> B using i
→ (Γ ⊢i A ---> B using i) *
(Γ ⊢i B ---> A using i)) *
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i)
→ Γ ⊢i A <---> B using i)
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ (Γ ⊢i A <---> B using i
→ (Γ ⊢i A ---> B using i) *
(Γ ⊢i B ---> A using i)) *
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i)
→ Γ ⊢i A <---> B using i)
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B
((Γ ⊢i A <---> B using i
→ (Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i)) *
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i)
→ Γ ⊢i A <---> B using i))%type
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A <---> B using i
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: ((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
Γ ⊢i A <---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A <---> B using i
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A <---> B using i H1: Γ ⊢i A ---> B using i
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A <---> B using i H1: Γ ⊢i A ---> B using i H2: Γ ⊢i B ---> A using i
((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
split; assumption.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: ((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
Γ ⊢i A <---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: ((Γ ⊢i A ---> B using i) * (Γ ⊢i B ---> A using i))%type
Γ ⊢i A <---> B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H1: Γ ⊢i A ---> B using i H2: Γ ⊢i B ---> A using i
Γ ⊢i A <---> B using i
apply pf_iff_split; assumption.}Defined.
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A <---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern
well_formed A → Γ ⊢i A <---> A using BasicReasoning
Σ: Signature Γ: Theory A: Pattern WFA: well_formed A
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A <---> B using i
→ Γ ⊢i B <---> C using i
→ Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A <---> B using i
→ Γ ⊢i B <---> C using i
→ Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AeqB: Γ ⊢i A <---> B using i BeqC: Γ ⊢i B <---> C using i
Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AeqB: ((Γ ⊢i A ---> B using i) *
(Γ ⊢i B ---> A using i))%type BeqC: Γ ⊢i B <---> C using i
Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BeqC: Γ ⊢i B <---> C using i
Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BeqC: ((Γ ⊢i B ---> C using i) *
(Γ ⊢i C ---> B using i))%type
Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i A <---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
((Γ ⊢i A ---> C using i) * (Γ ⊢i C ---> A using i))%type
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i C ---> A using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i A ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed ?B
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed C
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i A ---> ?Busing i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i ?B ---> C using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed B
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed C
1-3: wf_auto2.
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i C ---> A using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i C ---> A using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed C
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed ?B
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed A
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i C ---> ?Busing i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
Γ ⊢i ?B ---> A using i
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed C
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed B
Σ: Signature Γ: Theory A, B, C: Pattern i: ProofInfo wfA: well_formed A wfB: well_formed B wfC: well_formed C AimpB: Γ ⊢i A ---> B using i BimpA: Γ ⊢i B ---> A using i BimpC: Γ ⊢i B ---> C using i CimpB: Γ ⊢i C ---> B using i
well_formed A
1-3: wf_auto2.}Defined.
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Γ ⊢i b using i → Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Γ ⊢i b using i → Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b H: Γ ⊢i b using i
Γ ⊢i a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b H: Γ ⊢i b using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b H: Γ ⊢i b using i
Γ ⊢i ?ϕ₁ ---> a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b H: Γ ⊢i b using i
Γ ⊢i b ---> a ---> b using i
Σ: Signature Γ: Theory a, b: Pattern i: ProofInfo WFa: well_formed a WFb: well_formed b H: Γ ⊢i b using i
well_formed A
→ well_formed B
→ Γ ⊢i A using i → Γ ⊢i B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A using i → Γ ⊢i B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁ ---> B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
exact H.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i A ---> B ---> A and B using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i A ---> B ---> A and B using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed A
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed A
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed B
wf_auto2.}Defined.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A using i
→ Γ ⊢i A and B ---> C using i
→ Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A using i
→ Γ ⊢i A and B ---> C using i
→ Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed (A and B)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i B ---> A and B using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A and B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed B
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed (A and B)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i B ---> A and B using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A and B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed (A and B)
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i B ---> A and B using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A and B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed C
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i B ---> A and B using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A and B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A and B ---> C using i
exact H0.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i B ---> A and B using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed A
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed A
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
well_formed B
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i A and B ---> C using i
Γ ⊢i A using i
exact H.Defined.
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern
well_formed A
→ well_formed B
→ Γ ⊢i A ---> B ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i A ---> B ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (B and A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i B ---> A ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (B and A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i B ---> A ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (B and A)
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i B ---> A ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed (B and A)
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
Γ ⊢i B ---> A ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed B
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
Σ: Signature Γ: Theory A, B: Pattern WFA: well_formed A WFB: well_formed B
well_formed A
wf_auto2.}Defined.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A using i → Γ ⊢i B ---> B and A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A using i → Γ ⊢i B ---> B and A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i B ---> B and A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁ ---> B ---> B and A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i ?ϕ₁using i
exact H.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i A ---> B ---> B and A using i
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
Γ ⊢i A ---> B ---> B and A using BasicReasoning
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed A
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed A
wf_auto2.
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed B
Σ: Signature Γ: Theory A, B: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A using i
well_formed B
wf_auto2.}Defined.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A using i
→ Γ ⊢i B and A ---> C using i
→ Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A using i
→ Γ ⊢i B and A ---> C using i
→ Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed (B and A)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B ---> B and A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B and A ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed B
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed (B and A)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B ---> B and A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B and A ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed (B and A)
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B ---> B and A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B and A ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed C
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B ---> B and A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B and A ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i B ---> B and A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed A
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed A
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i A using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
well_formed B
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A using i H0: Γ ⊢i B and A ---> C using i
Γ ⊢i A using i
exact H.Defined.
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A and B using i → Γ ⊢i B and A using i
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ Γ ⊢i A and B using i → Γ ⊢i B and A using i
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
Γ ⊢i B and A using i
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i P1: Γ ⊢i B using i
Γ ⊢i B and A using i
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed A
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed B
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i P1: Γ ⊢i B using i P2: Γ ⊢i A using i
Γ ⊢i B and A using i
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i P1: Γ ⊢i B using i
well_formed A
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i P1: Γ ⊢i B using i
well_formed B
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed A
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i
well_formed B
Σ: Signature A, B: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B H: Γ ⊢i A and B using i P1: Γ ⊢i B using i P2: Γ ⊢i A using i
Γ ⊢i B and A using i
apply conj_intro_meta; assumption.Defined.
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo
Γ ⊢i r' ---> r using i
→ ∀l : hypotheses,
Γ Ⱶ
l--------------------------------------
r'
using i
→ Γ Ⱶ
l--------------------------------------
r
using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo
Γ ⊢i r' ---> r using i
→ ∀l : hypotheses,
Γ Ⱶ
l--------------------------------------
r'
using i
→ Γ Ⱶ
l--------------------------------------
r
using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: Γ Ⱶ
l--------------------------------------
r'
using i
Γ Ⱶ
l--------------------------------------
r
using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
r'
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
r'
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
r'
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
r'
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
r'
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
r'
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
r
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
r
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
r
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
r
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
r
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
r
using i
)
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i
well_formed r
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r (patterns_of l) using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp r (patterns_of l) using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed ?g
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> r using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp ?g (patterns_of l) using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r'
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp r' (patterns_of l) using i
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r'
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r'
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
well_formed r'
Σ: Signature Γ: Theory r, r': Pattern i: ProofInfo Himp: Γ ⊢i r' ---> r using i l: hypotheses H: well_formed r'
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp r' (patterns_of l) using i wfr: well_formed r wfl: Pattern.wf (patterns_of l)
Γ Ⱶ
l--------------------------------------
x
using i
→ Γ Ⱶ
l--------------------------------------
x or y
using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: Γ Ⱶ
l--------------------------------------
x
using i
Γ Ⱶ
l--------------------------------------
x or y
using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
x
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
x
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
x
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
x
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
x
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
x
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
x or y
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
x or y
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
x or y
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
x or y
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
x or y
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
x or y
using i
)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i
well_formed (x or y)
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp (x or y) (patterns_of l)
using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (x or y) (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed ?g
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> x or y using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp ?g (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> x or y using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> x or y using BasicReasoning
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp x (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp x (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp x (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp x (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed x
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp x (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ Ⱶ
l--------------------------------------
y
using i
→ Γ Ⱶ
l--------------------------------------
x or y
using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: Γ Ⱶ
l--------------------------------------
y
using i
Γ Ⱶ
l--------------------------------------
x or y
using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
y
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
y
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
y
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
y
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
y
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
y
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
x or y
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
x or y
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
x or y
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
x or y
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
x or y
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
x or y
using i
)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i
well_formed (x or y)
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp (x or y) (patterns_of l)
using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (x or y) (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed ?g
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> x or y using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp ?g (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> x or y using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i ?g ---> x or y using BasicReasoning
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed x
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp y (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp y (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp y (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed (x or y)
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp y (patterns_of l) using i
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
well_formed y
wf_auto2.
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory l: hypotheses x, y: Pattern i: ProofInfo H: well_formed y
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp y (patterns_of l) using i wfxy: well_formed (x or y) wfl: Pattern.wf (patterns_of l)
Γ ⊢i r ---> r' using i
→ ∀ (l₁l₂ : list named_hypothesis) (g : Pattern),
Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
→ Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
Γ ⊢i r ---> r' using i
→ ∀ (l₁l₂ : list named_hypothesis) (g : Pattern),
Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
→ Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern H: Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern H: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (n' ∶ r') :: l₂--------------------------------------
g
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (n ∶ r) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern H: well_formed g
→ Pattern.wf (patterns_of (l₁ ++ (n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l₁ ++ (n' ∶ r') :: l₂))
using i
well_formed g
→ Pattern.wf (patterns_of (l₁ ++ (n ∶ r) :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l₁ ++ (n ∶ r) :: l₂)) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern H: well_formed g
→ Pattern.wf (patterns_of (l₁ ++ (n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l₁ ++ (n' ∶ r') :: l₂))
using i wfg: well_formed g Hwf: Pattern.wf (patterns_of (l₁ ++ (n ∶ r) :: l₂))
Γ
⊢i foldr patt_imp g
(patterns_of (l₁ ++ (n ∶ r) :: l₂)) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern H: Pattern.wf (patterns_of (l₁ ++ (n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(patterns_of (l₁ ++ (n' ∶ r') :: l₂))
using i wfg: well_formed g Hwf: Pattern.wf (patterns_of (l₁ ++ (n ∶ r) :: l₂))
Γ
⊢i foldr patt_imp g
(patterns_of (l₁ ++ (n ∶ r) :: l₂)) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern H: Pattern.wf (map nh_patt (l₁ ++ (n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt (l₁ ++ (n' ∶ r') :: l₂))
using i wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂))
Γ
⊢i foldr patt_imp g
(map nh_patt (l₁ ++ (n ∶ r) :: l₂)) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Γ
⊢i foldr patt_imp g
(map nh_patt (l₁ ++ (n ∶ r) :: l₂)) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++ map nh_patt ((n ∶ r) :: l₂))
using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Pattern.wf (map nh_patt l₁)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed ?h
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed (nh_patt (n ∶ r))
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed g
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Γ ⊢i nh_patt (n ∶ r) ---> ?husing i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
?h
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Pattern.wf (map nh_patt l₁)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed r'
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed (nh_patt (n ∶ r))
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed g
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
r'
:: (fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂) using i
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Pattern.wf (map nh_patt l₁)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
Pattern.wf
((fix map (l : list named_hypothesis) :
list Pattern :=
match l with
| [] => []
| a :: t => nh_patt a :: map t
end) l₂)
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed r'
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed (nh_patt (n ∶ r))
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed g
Σ: Signature Γ: Theory n: string r: Pattern n': string r': Pattern i: ProofInfo Himp: Γ ⊢i r ---> r' using i l₁, l₂: list named_hypothesis g: Pattern wfg: well_formed g Hwf: Pattern.wf (map nh_patt (l₁ ++ (n ∶ r) :: l₂)) H: Pattern.wf
(map nh_patt l₁ ++ map nh_patt ((n' ∶ r') :: l₂))
→ Γ
⊢i foldr patt_imp g
(map nh_patt l₁ ++
map nh_patt ((n' ∶ r') :: l₂)) using i
well_formed p
→ well_formed q
→ Γ ⊢i p ---> p or q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern
well_formed p
→ well_formed q
→ Γ ⊢i p ---> p or q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ ⊢i p ---> p or q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
well_formed (p ---> p or q)
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
p ---> p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
well_formed (p ---> p or q)
wf_auto2.
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
p ---> p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ p,
--------------------------------------
p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ p or q,
--------------------------------------
p or q
using BasicReasoning
mlExact "H0".Defined.(* All thic complicated code is here only for one reason: I want to be able to first run the tactic with all the parameters inferred or question marked, which results in another goal. And then I want to handle the subgoals generated by the filling-in underscores / question marks. In particular, I want the proof mode goals to be generated first, and the other, uninteresting goals, next.*)Ltac2 rec fillWithUnderscoresAndCall (tac : constr -> unit) (t : constr) :=
lazy_match! Constr.type t with
| (?t' -> ?t's) =>
lazy_match! goal with
| [|- ?g] =>
leth := Fresh.in_goal ident:(h) inassert(h : $t' -> $g) > [(
letpftprime := Fresh.in_goal ident:(pftprime) inintro $pftprime;
letnew_t := open_constr:($t ltac2:(Notations.exact0 false (fun () => Control.hyp (pftprime)))) in
fillWithUnderscoresAndCall tac new_t;
Std.clear [pftprime]
)|(apply &h)
]
end
| (forall_ : _, _) => fillWithUnderscoresAndCall tac open_constr:($t _)
| _ => tac t
end
.(* We have this double-primed version because we want to be able to feed it the proof before feeding it the ProofInfoLe.*)
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo
Γ ⊢i ϕ using i' → ProofInfoLe i' i → Γ ⊢i ϕ using i
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo
Γ ⊢i ϕ using i' → ProofInfoLe i' i → Γ ⊢i ϕ using i
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo H: Γ ⊢i ϕ using i' pile: ProofInfoLe i' i
Γ ⊢i ϕ using i
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo H: Γ ⊢i ϕ using i' pile: ProofInfoLe i' i
ProofInfoLe ?i' i
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo H: Γ ⊢i ϕ using i' pile: ProofInfoLe i' i
Γ ⊢i ϕ using?i'
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo H: Γ ⊢i ϕ using i' pile: ProofInfoLe i' i
ProofInfoLe ?i' i
apply pile.
Σ: Signature Γ: Theory ϕ: Pattern i', i: ProofInfo H: Γ ⊢i ϕ using i' pile: ProofInfoLe i' i
Γ ⊢i ϕ using i'
apply H.Defined.Ltac2 _callCompletedAndCast (t : constr) (tac : constr -> unit) :=
lettac' := (fun (t' : constr) =>
lettcast := open_constr:(@useGenericReasoning'' _ _ _ _ _ $t') in
fillWithUnderscoresAndCall tac tcast
) in
fillWithUnderscoresAndCall tac' t
.Ltac2 try_solve_pile_basic () :=
Control.enter (fun () =>
lazy_match! goal with
| [ |- (@ProofInfoLe _ _ _)] =>
try (solve
[ apply pile_any
| apply pile_refl
| apply pile_basic_generic'
])
| [|- _] => ()
end
)
.Ltac2 try_wfa () :=
Control.enter (fun () =>
letwfa := (funp =>
if (Constr.has_evar p) then
()
else
ltac1:(try_wfauto2)
) inlazy_match! goal with
| [|- well_formed ?p = true] => wfa p
| [|- is_true (well_formed ?p) ] => wfa p
| [|- _] => ()
end
)
.Ltac2 mlApplyMeta (t : constr) :=
_callCompletedAndCast t _mlApplyMetaRaw ;
try_solve_pile_basic ();
try_wfa ()
.Ltac2 mlApplyMetaIn (t : constr) (name : constr) :=
_callCompletedAndCast t (funt =>
_mlApplyMetaRawIn t name
);
try_solve_pile_basic ();
try_wfa ()
.Ltac_mlApplyMeta t :=
letff := ltac2:(t' |- mlApplyMeta (Option.get (Ltac1.to_constr(t')))) in
ff t.Ltac_mlApplyMetaIn t name :=
letff := ltac2:(t' name' |- mlApplyMetaIn (Option.get (Ltac1.to_constr(t'))) (Option.get (Ltac1.to_constr(name')))) in
ff t name
.Tactic Notation"mlApplyMeta"constr(t) :=
_mlApplyMeta t.Tactic Notation"mlApplyMeta"constr(t) "in"constr(name) :=
_mlApplyMetaIn t name
.
Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
→ Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
→ Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))) wfg': well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))) wfg': well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf': Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
→ well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
→ well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
→ mlTheory
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)
auto.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed
(mlConclusion
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)) Hwf: Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
)))
Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf: Pattern.wf (patterns_of (l₁ ++ (nxy ∶ x and y) :: l₂))
Γ Ⱶ
l₁ ++ (nxy ∶ x and y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf: Pattern.wf (patterns_of (l₁ ++ (nxy ∶ x and y) :: l₂)) l1: list named_hypothesis Heql1: l1 =
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) l2: list named_hypothesis Heql2: l2 =
drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)
well_formed y
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf: Pattern.wf
(patterns_of (l₁ ++ (nxy ∶ x and y) :: l₂)) Hwf0: well_formed y
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf: Pattern.wf
(patterns_of (l₁ ++ (nxy ∶ x and y) :: l₂)) Hwf0: well_formed y
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g l1: list named_hypothesis Heql1: l1 =
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) l2: list named_hypothesis Heql2: l2 =
drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
well_formed y
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g l1: list named_hypothesis Heql1: l1 =
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) l2: list named_hypothesis Heql2: l2 =
drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
well_formed y
abstract (
apply wfapp_proj_2 in Hwf;
unfold Pattern.wfin Hwf;
simplin Hwf;
apply andb_prop in Hwf;
destruct Hwf as [wfxy _];
wf_auto2
).
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
))) (l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
))) (l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
?hyps--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
l₁ ++ (nxy ∶ x and y,
) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of (l₁ ++ (nxy ∶ x and y,
))
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
well_formed x
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) H0: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
well_formed x
abstract (
apply wfapp_proj_2 in Hwf;
unfold Pattern.wfin Hwf;
simplin Hwf;
apply andb_prop in Hwf as [wfxy _];
wf_auto2
).
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) H0: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) H0: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
?Aand y
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) H0: well_formed x h: well_formed ?A
→ Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
y
using i
well_formed ?A
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) H0: well_formed x h: well_formed x
→ Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
y
using i
well_formed x
wf_auto2.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
))) (l₁ ++ (nxy ∶ x and y) :: l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
?hyps--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
(l₁ ++ (nxy ∶ x and y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
l₁ ++ (nxy ∶ x and y,
) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
((l₁ ++ (nxy ∶ x and y,
)) ++
(ny ∶ y)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
patterns_of ?hyps =
patterns_of
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂))
Γ Ⱶ
(l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) l1: list named_hypothesis Heql1: l1 =
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) l2: list named_hypothesis Heql2: l2 =
drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)
well_formed x
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) l1: list named_hypothesis Heql1: l1 =
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) l2: list named_hypothesis Heql2: l2 =
drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)
well_formed x
abstract (
apply wfapp_proj_2 in Hwf;
unfold Pattern.wfin Hwf;
simplin Hwf;
apply andb_prop in Hwf;
destruct Hwf as [wfxy _];
wf_auto2
).
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
?hyps--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
l₁ ++ (nxy ∶ x and y,
) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of (l₁ ++ (nxy ∶ x and y,
))
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
well_formed x
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1, H0: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
well_formed x
abstract (
apply wfapp_proj_2 in Hwf;
unfold Pattern.wfin Hwf;
simplin Hwf;
apply andb_prop in Hwf as [wfxy _];
wf_auto2
).
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1, H0: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
x
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1, H0: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
x and?Busing i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1, H0: well_formed x h: well_formed ?B
→ Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
x
using i
well_formed ?B
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1, H0: well_formed x h: well_formed y
→ Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
)--------------------------------------
x
using i
well_formed y
assumption.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂)--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
?hyps--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
(l₁ ++ (nxy ∶ x and y,
)) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
l₁ ++ (nxy ∶ x and y,
) ++ l₂ = l₁ ++ (nxy ∶ x and y) :: l₂
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(take (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
((l₁ ++ (nxy ∶ x and y,
)) ++
(nx ∶ x)
:: drop (length (l₁ ++ (nxy ∶ x and y,
)))
((l₁ ++ (nxy ∶ x and y,
)) ++ (ny ∶ y) :: l₂))
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
((l₁ ++ (nxy ∶ x and y,
)) ++ (nx ∶ x) :: (ny ∶ y) :: l₂)
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
(l₁ ++ (nxy ∶ x and y,
)) ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
((l₁ ++ (nxy ∶ x and y,
)) ++ (nx ∶ x) :: (ny ∶ y) :: l₂)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
?hyps--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
((l₁ ++ (nxy ∶ x and y,
)) ++ (nx ∶ x) :: (ny ∶ y) :: l₂)
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
patterns_of ?hyps =
patterns_of
(l₁ ++
(nxy ∶ x and y,
) ++ (nx ∶ x) :: (ny ∶ y) :: l₂)
reflexivity.
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
l₁ ++ (nxy ∶ x and y,
) ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
Σ: Signature Γ: Theory g: Pattern l₁, l₂: list named_hypothesis nx: string x: Pattern ny: string y: Pattern nxy: string i: ProofInfo H: Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
wfg: well_formed g Hwf0: well_formed y Hwf: Pattern.wf
(map nh_patt l₁ ++
map nh_patt ((nxy ∶ x and y) :: l₂)) Hwf1: well_formed x
Γ Ⱶ
l₁ ++ (nx ∶ x) :: (ny ∶ y) :: l₂--------------------------------------
g
using i
well_formed a
→ well_formed b
→ well_formed p
→ well_formed q
→ Γ ⊢i p and q ---> a and b ---> q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern
well_formed a
→ well_formed b
→ well_formed p
→ well_formed q
→ Γ ⊢i p and q ---> a and b ---> q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ ⊢i p and q ---> a and b ---> q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
well_formed (p and q ---> a and b ---> q ---> a)
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
p and q ---> a and b ---> q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
well_formed (p and q ---> a and b ---> q ---> a)
wf_auto2.
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
p and q ---> a and b ---> q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
"H0" ∶ p and q,
--------------------------------------
a and b ---> q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
"H0" ∶ p and q,
"H1" ∶ a and b,
--------------------------------------
q ---> a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
"H0" ∶ p and q,
"H1" ∶ a and b,
"H2" ∶ q,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
"H0" ∶ p and q,
"H3" ∶ a,
"H4" ∶ b,
"H2" ∶ q,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a, b, p, q: Pattern H: well_formed a H0: well_formed b H1: well_formed p H2: well_formed q
Γ Ⱶ
"0" ∶ p,
"1" ∶ q,
"H3" ∶ a,
"H4" ∶ b,
"H2" ∶ q,
--------------------------------------
a
using BasicReasoning
mlExact "H3".Defined.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p <---> p' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p <---> p' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q <---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q <---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p and q ---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p' and q' ---> p and q using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p and q ---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p and q ---> p' and q')
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p and q ---> p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p and q ---> p' and q')
wf_auto2.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p and q ---> p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p and q,
--------------------------------------
p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! (! p or ! q),
--------------------------------------
! (! p' or ! q')
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! (! p or ! q),
"H1" ∶ ! p' or ! q',
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H1" ∶ ! p' or ! q',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H2" ∶ ! p',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H3" ∶ ! q',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H2" ∶ ! p',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i ! p' ---> ! p using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H2" ∶ ! p',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i ! p' ---> ! p using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! p' ---> ! p,
"H0" ∶ ! p or ! q ---> ⊥,
"H2" ∶ ! p',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i ! p' ---> ! p using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! p' ---> ! p,
"H0" ∶ ! p or ! q ---> ⊥,
"H2" ∶ ! p',
--------------------------------------
! p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i ! p' ---> ! p using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! p' ---> ! p,
"H0" ∶ ! p or ! q ---> ⊥,
"H2" ∶ ! p',
--------------------------------------
! p'
using i
mlExact "H2".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H3" ∶ ! q',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i ! q' ---> ! q using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p or ! q ---> ⊥,
"H3" ∶ ! q',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i ! q' ---> ! q using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! q' ---> ! q,
"H0" ∶ ! p or ! q ---> ⊥,
"H3" ∶ ! q',
--------------------------------------
! p or ! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i ! q' ---> ! q using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! q' ---> ! q,
"H0" ∶ ! p or ! q ---> ⊥,
"H3" ∶ ! q',
--------------------------------------
! q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i ! q' ---> ! q using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! q' ---> ! q,
"H0" ∶ ! p or ! q ---> ⊥,
"H3" ∶ ! q',
--------------------------------------
! q'
using i
mlExact "H3".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p' and q' ---> p and q using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p' and q' ---> p and q)
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' and q' ---> p and q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p' and q' ---> p and q)
wf_auto2.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' and q' ---> p and q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
--------------------------------------
p and q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! (! p' or ! q'),
--------------------------------------
! (! p or ! q)
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! (! p' or ! q'),
"H1" ∶ ! p or ! q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H1" ∶ ! p or ! q,
--------------------------------------
! p' or ! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H2" ∶ ! p,
--------------------------------------
! p' or ! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H3" ∶ ! q,
--------------------------------------
! p' or ! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H2" ∶ ! p,
--------------------------------------
! p' or ! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H2" ∶ ! p,
--------------------------------------
! p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i ! p ---> ! p' using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H2" ∶ ! p,
--------------------------------------
! p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i ! p ---> ! p' using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! p ---> ! p',
"H0" ∶ ! p' or ! q' ---> ⊥,
"H2" ∶ ! p,
--------------------------------------
! p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i ! p ---> ! p' using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ ! p ---> ! p',
"H0" ∶ ! p' or ! q' ---> ⊥,
"H2" ∶ ! p,
--------------------------------------
! p
using i
mlExact "H2".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H3" ∶ ! q,
--------------------------------------
! p' or ! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H3" ∶ ! q,
--------------------------------------
! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i ! q ---> ! q' using i
Γ Ⱶ
"H0" ∶ ! p' or ! q' ---> ⊥,
"H3" ∶ ! q,
--------------------------------------
! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i ! q ---> ! q' using i
Γ Ⱶ
"H1" ∶ ! q ---> ! q',
"H0" ∶ ! p' or ! q' ---> ⊥,
"H3" ∶ ! q,
--------------------------------------
! q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i ! q ---> ! q' using i
Γ Ⱶ
"H1" ∶ ! q ---> ! q',
"H0" ∶ ! p' or ! q' ---> ⊥,
"H3" ∶ ! q,
--------------------------------------
! q
using i
mlExact "H3".Defined.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p <---> p' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p <---> p' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q <---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q <---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p or q ---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p' or q' ---> p or q using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p or q ---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p or q ---> p' or q')
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p or q ---> p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p or q ---> p' or q')
auto.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p or q ---> p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p or q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p ---> p' using i
assumption.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i q ---> q' using i
assumption.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p' or q' ---> p or q using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p' or q' ---> p or q)
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' or q' ---> p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p' or q' ---> p or q)
auto.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' or q' ---> p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' or q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p' ---> p using i
assumption.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i q' ---> q using i
assumption.Defined.
Σ: Signature Γ: Theory p, q: Pattern
well_formed p
→ well_formed q
→ Γ ⊢i (p ---> q) <---> ! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern
well_formed p
→ well_formed q
→ Γ ⊢i (p ---> q) <---> ! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ ⊢i (p ---> q) <---> ! p or q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ ⊢i (p ---> q) ---> ! p or q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ ⊢i ! p or q ---> p ---> q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ ⊢i (p ---> q) ---> ! p or q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
well_formed ((p ---> q) ---> ! p or q)
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
(p ---> q) ---> ! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
well_formed ((p ---> q) ---> ! p or q)
wf_auto2.
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
(p ---> q) ---> ! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ p ---> q,
--------------------------------------
! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H1" ∶ p or ! p,
"H0" ∶ p ---> q,
--------------------------------------
! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H2" ∶ p,
"H0" ∶ p ---> q,
--------------------------------------
! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H3" ∶ ! p,
"H0" ∶ p ---> q,
--------------------------------------
! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H2" ∶ p,
"H0" ∶ p ---> q,
--------------------------------------
! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H2" ∶ p,
"H0" ∶ p ---> q,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H2" ∶ p,
"H0" ∶ p ---> q,
--------------------------------------
p
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H3" ∶ ! p,
"H0" ∶ p ---> q,
--------------------------------------
! p or q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H3" ∶ ! p,
"H0" ∶ p ---> q,
--------------------------------------
! p
using BasicReasoning
mlExact "H3".
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ ⊢i ! p or q ---> p ---> q using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
well_formed (! p or q ---> p ---> q)
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
! p or q ---> p ---> q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
well_formed (! p or q ---> p ---> q)
wf_auto2.
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
! p or q ---> p ---> q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ ! p or q,
--------------------------------------
p ---> q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ ! p or q,
"H2" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ ! ! p ---> q,
"H2" ∶ p,
--------------------------------------
q
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ ! ! p ---> q,
"H2" ∶ p,
--------------------------------------
! ! p
using BasicReasoning
Σ: Signature Γ: Theory p, q: Pattern wfp: well_formed p wfq: well_formed q
Γ Ⱶ
"H0" ∶ ! ! p ---> q,
"H2" ∶ p,
--------------------------------------
p
using BasicReasoning
mlExact "H2".Defined.
Σ: Signature Γ: Theory p: Pattern
well_formed p
→ Γ ⊢i ⊥ <---> p and ! p using BasicReasoning
Σ: Signature Γ: Theory p: Pattern
well_formed p
→ Γ ⊢i ⊥ <---> p and ! p using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ ⊢i ⊥ <---> p and ! p using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ ⊢i ⊥ ---> p and ! p using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ ⊢i p and ! p ---> ⊥ using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ ⊢i ⊥ ---> p and ! p using BasicReasoning
apply bot_elim; auto.
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ ⊢i p and ! p ---> ⊥ using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ ⊢i ! (! p or ! ! p) ---> ⊥ using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
well_formed (! (! p or ! ! p) ---> ⊥)
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ Ⱶ
! (! p or ! ! p) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
well_formed (! (! p or ! ! p) ---> ⊥)
wf_auto2.
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ Ⱶ
! (! p or ! ! p) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ Ⱶ
"H0" ∶ ! (! p or ! ! p),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ Ⱶ
"H0" ∶ ! p or ! ! p ---> ⊥,
--------------------------------------
! p or ! ! p
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern wfp: well_formed p
Γ Ⱶ
"H1" ∶ ! p or ! ! p,
"H0" ∶ ! p or ! ! p ---> ⊥,
--------------------------------------
! p or ! ! p
using BasicReasoning
well_formed A
→ well_formed B
→ well_formed L
→ well_formed R
→ Γ
⊢i (L and A ---> B or R) --->
L ---> (A ---> B) or R using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern
well_formed A
→ well_formed B
→ well_formed L
→ well_formed R
→ Γ
⊢i (L and A ---> B or R) --->
L ---> (A ---> B) or R using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ ⊢i (L and A ---> B or R) ---> L ---> (A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
well_formed
((L and A ---> B or R) ---> L ---> (A ---> B) or R)
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
(L and A ---> B or R) ---> L ---> (A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
well_formed
((L and A ---> B or R) ---> L ---> (A ---> B) or R)
wf_auto2.
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
(L and A ---> B or R) ---> L ---> (A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H0" ∶ L and A ---> B or R,
--------------------------------------
L ---> (A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H2" ∶ A or ! A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H4" ∶ ! A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
well_formed (B or R)
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
B or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ B or R,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
well_formed (B or R)
wf_auto2.
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
B or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ B or R,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
B or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
L and A
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
! (! L or ! A)
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ ! L or ! A,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ ! L,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H5" ∶ ! A,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ ! L,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ L ---> ⊥,
--------------------------------------
L
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H5" ∶ ! A,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H5" ∶ A ---> ⊥,
--------------------------------------
A
using BasicReasoning
mlExact "H3".
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ B or R,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ B,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H5" ∶ R,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ B,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ B,
--------------------------------------
A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H4" ∶ B,
"H2" ∶ A,
--------------------------------------
B
using BasicReasoning
mlExact "H4".
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H5" ∶ R,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R Hwf: well_formed (B or R)
Γ Ⱶ
"H3" ∶ A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H5" ∶ R,
--------------------------------------
R
using BasicReasoning
mlExact "H5".
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H4" ∶ ! A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
(A ---> B) or R
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H4" ∶ ! A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
--------------------------------------
A ---> B
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H4" ∶ ! A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ A,
--------------------------------------
B
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H4" ∶ ! A,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ A,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory A, B, L, R: Pattern wfA: well_formed A wfB: well_formed B wfL: well_formed L wfR: well_formed R
Γ Ⱶ
"H4" ∶ A ---> ⊥,
"H0" ∶ L and A ---> B or R,
"H1" ∶ L,
"H2" ∶ A,
--------------------------------------
A
using BasicReasoning
mlExact "H2".Defined.
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed L
→ well_formed R
→ Γ ⊢i L and A ---> B or R using i
→ Γ ⊢i L ---> (A ---> B) or R using i
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed L
→ well_formed R
→ Γ ⊢i L and A ---> B or R using i
→ Γ ⊢i L ---> (A ---> B) or R using i
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFL: well_formed L WFR: well_formed R H: Γ ⊢i L and A ---> B or R using i
Γ ⊢i L ---> (A ---> B) or R using i
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFL: well_formed L WFR: well_formed R H: Γ ⊢i L and A ---> B or R using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFL: well_formed L WFR: well_formed R H: Γ ⊢i L and A ---> B or R using i
Γ ⊢i ?ϕ₁ ---> L ---> (A ---> B) or R using i
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFL: well_formed L WFR: well_formed R H: Γ ⊢i L and A ---> B or R using i
Γ ⊢i ?ϕ₁ ---> L ---> (A ---> B) or R using i
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFL: well_formed L WFR: well_formed R H: Γ ⊢i L and A ---> B or R using i
Γ ⊢i ?ϕ₁ ---> L ---> (A ---> B) or R
using BasicReasoning
apply weird_lemma; assumption.
Σ: Signature Γ: Theory A, B, L, R: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFL: well_formed L WFR: well_formed R H: Γ ⊢i L and A ---> B or R using i
Γ ⊢i L and A ---> B or R using i
exact H.Defined.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ well_formed D
→ Γ ⊢i C ---> A using i
→ Γ ⊢i B ---> D using i
→ Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ well_formed D
→ Γ ⊢i C ---> A using i
→ Γ ⊢i B ---> D using i
→ Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (B ---> D) ---> (A ---> B) ---> A ---> D
using BasicReasoning
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (B ---> D) ---> (A ---> B) ---> A ---> D
using?i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using?i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (B ---> D) ---> (A ---> B) ---> A ---> D
using?i
Γ ⊢i B ---> D using?i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (B ---> D) ---> (A ---> B) ---> A ---> D
using?i
Γ ⊢i B ---> D using?i
exact H0.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (C ---> A) ---> (A ---> D) ---> C ---> D
using BasicReasoning
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (C ---> A) ---> (A ---> D) ---> C ---> D
using?i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (A ---> D) ---> C ---> D using?i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (C ---> A) ---> (A ---> D) ---> C ---> D
using?i
Γ ⊢i C ---> A using?i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (C ---> A) ---> (A ---> D) ---> C ---> D
using?i
Γ ⊢i C ---> A using?i
exact H.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (A ---> D) ---> C ---> D using i
Γ ⊢i (A ---> B) ---> C ---> D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (A ---> D) ---> C ---> D using i H3: Γ ⊢i (A ---> B) ---> C ---> D using i
Γ ⊢i (A ---> B) ---> C ---> D using i
exact H3.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (A ---> D) ---> C ---> D using i
well_formed (A ---> B)
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (A ---> D) ---> C ---> D using i
well_formed (A ---> D)
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i C ---> A using i H0: Γ ⊢i B ---> D using i H1: Γ ⊢i (A ---> B) ---> A ---> D using i H2: Γ ⊢i (A ---> D) ---> C ---> D using i
well_formed (C ---> D)
all: wf_auto2.Defined.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i B ---> C using i
→ Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i B ---> C using i
→ Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i
Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁ ---> A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁ ---> A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁ ---> A and B ---> A and C
using BasicReasoning
exact H0.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i A ---> B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
well_formed A
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
well_formed (A and C)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i B ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
Γ ⊢i B ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i B ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁ ---> B ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁0using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁ ---> B ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁ ---> B ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i ?ϕ₁0 ---> ?ϕ₁ ---> B ---> A ---> A and C
using BasicReasoning
exact H1.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i C ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i B ---> C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i B ---> C using i
exact H.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i C ---> A ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning H1: Γ
⊢i (B ---> C) --->
(C ---> A ---> A and C) --->
B ---> A ---> A and C using BasicReasoning
Γ ⊢i C ---> A ---> A and C using BasicReasoning
apply conj_intro2; assumption.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i
well_formed A
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i
well_formed (A and C)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
well_formed C
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
well_formed B
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i B ---> C using i H0: Γ
⊢i (A ---> B ---> A and C) --->
A and B ---> A and C using BasicReasoning
well_formed (A ---> A and C)
all: wf_auto2.Defined.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ well_formed D
→ Γ ⊢i A ---> B using i
→ Γ ⊢i C ---> D using i
→ Γ ⊢i A and C ---> B and D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ well_formed D
→ Γ ⊢i A ---> B using i
→ Γ ⊢i C ---> D using i
→ Γ ⊢i A and C ---> B and D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ ⊢i A and C ---> B and D using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
well_formed (A and C ---> B and D)
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
A and C ---> B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
well_formed (A and C ---> B and D)
wf_auto2.
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
A and C ---> B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
A and C ---> B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H0" ∶ A ---> B,
--------------------------------------
A and C ---> B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
--------------------------------------
A and C ---> B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H2" ∶ A and C,
--------------------------------------
B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
--------------------------------------
B and D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H5" ∶ ! B or ! D,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H6" ∶ ! B,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H7" ∶ ! D,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H6" ∶ ! B,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H6" ∶ B ---> ⊥,
--------------------------------------
B
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H6" ∶ B ---> ⊥,
--------------------------------------
A
using i
mlExact "H3".
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H7" ∶ ! D,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H7" ∶ ! D,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H7" ∶ D ---> ⊥,
--------------------------------------
D
using i
Σ: Signature Γ: Theory A, B, C, D: Pattern i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C WFD: well_formed D H: Γ ⊢i A ---> B using i H0: Γ ⊢i C ---> D using i
Γ Ⱶ
"H1" ∶ C ---> D,
"H0" ∶ A ---> B,
"H3" ∶ A,
"H4" ∶ C,
"H7" ∶ D ---> ⊥,
--------------------------------------
C
using i
mlExact "H4".}}Defined.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A and B ---> C using i
→ Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo
well_formed A
→ well_formed B
→ well_formed C
→ Γ ⊢i A and B ---> C using i
→ Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ ⊢i A and B ---> A and C using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
well_formed (A and B ---> A and C)
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
A and B ---> A and C
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
well_formed (A and B ---> A and C)
wf_auto2.
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
A and B ---> A and C
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
--------------------------------------
A and B ---> A and C
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
--------------------------------------
A and C
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H2" ∶ ! A or ! C,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H3" ∶ ! A,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H4" ∶ ! C,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H3" ∶ ! A,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H5" ∶ A,
"H6" ∶ B,
"H3" ∶ ! A,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H5" ∶ A,
"H6" ∶ B,
"H3" ∶ A ---> ⊥,
--------------------------------------
A
using i
mlExact "H5".
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H4" ∶ ! C,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H4" ∶ ! C,
--------------------------------------
⊥
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H4" ∶ C ---> ⊥,
--------------------------------------
C
using i
Σ: Signature A, B, C: Pattern Γ: Theory i: ProofInfo WFA: well_formed A WFB: well_formed B WFC: well_formed C H: Γ ⊢i A and B ---> C using i
Γ Ⱶ
"H0" ∶ A and B ---> C,
"H1" ∶ A and B,
"H4" ∶ C ---> ⊥,
--------------------------------------
A and B
using i
mlExact "H1".}Defined.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo
well_formed a = true
→ well_formed b = true
→ well_formed a' = true
→ well_formed b' = true
→ Γ ⊢i a <---> a' using i
→ Γ ⊢i b <---> b' using i
→ Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo
well_formed a = true
→ well_formed b = true
→ well_formed a' = true
→ well_formed b' = true
→ Γ ⊢i a <---> a' using i
→ Γ ⊢i b <---> b' using i
→ Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i
well_formed (a ---> a')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i
well_formed (a' ---> a)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i
well_formed (a ---> a')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i
well_formed (a' ---> a)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i
well_formed (a' ---> a)
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
well_formed (a ---> a')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
well_formed (a' ---> a)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
well_formed (a ---> a')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
well_formed (a' ---> a)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i
well_formed (a' ---> a)
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
well_formed (b ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
well_formed (b' ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
well_formed (b ---> b')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
well_formed (b' ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i
well_formed (b' ---> b)
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
well_formed (b ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
well_formed (b' ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
well_formed (b ---> b')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
well_formed (b' ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i
well_formed (b' ---> b)
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed (a ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed (a ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed (a' ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) <---> (a ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) <---> (a ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) <---> (a ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b) ---> a ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b') ---> a ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) ---> a ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') ---> a ---> b using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) ---> a ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') ---> a ---> b using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b) ---> a ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b) ---> a ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a ---> b) ---> a ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b) ---> a ---> b')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a ---> b) ---> a ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a ---> b,
--------------------------------------
a ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a ---> b,
"H1" ∶ a,
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ b ---> b',
"H0" ∶ a ---> b,
"H1" ∶ a,
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ b ---> b',
"H0" ∶ a ---> b,
"H1" ∶ a,
--------------------------------------
b
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ b ---> b',
"H0" ∶ a ---> b,
"H1" ∶ a,
--------------------------------------
a
using i
mlExact "H1".
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') ---> a ---> b using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b') ---> a ---> b)
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a ---> b') ---> a ---> b
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b') ---> a ---> b)
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a ---> b') ---> a ---> b
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a ---> b',
--------------------------------------
a ---> b
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a ---> b',
"H1" ∶ a,
--------------------------------------
b
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ b' ---> b,
"H0" ∶ a ---> b',
"H1" ∶ a,
--------------------------------------
b
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ b' ---> b,
"H0" ∶ a ---> b',
"H1" ∶ a,
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ b' ---> b,
"H0" ∶ a ---> b',
"H1" ∶ a,
--------------------------------------
a
using i
mlExact "H1".
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') <---> (a' ---> b') using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b') ---> a' ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a' ---> b') ---> a ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') ---> a' ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a' ---> b') ---> a ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') ---> a' ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a' ---> b') ---> a ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a ---> b') ---> a' ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b') ---> a' ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a ---> b') ---> a' ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a ---> b') ---> a' ---> b')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a ---> b') ---> a' ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a ---> b',
--------------------------------------
a' ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a ---> b',
"H1" ∶ a',
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ a' ---> a,
"H0" ∶ a ---> b',
"H1" ∶ a',
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ a' ---> a,
"H0" ∶ a ---> b',
"H1" ∶ a',
--------------------------------------
a
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ a' ---> a,
"H0" ∶ a ---> b',
"H1" ∶ a',
--------------------------------------
a'
using i
mlExact "H1".
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ ⊢i (a' ---> b') ---> a ---> b' using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a' ---> b') ---> a ---> b')
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a' ---> b') ---> a ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
well_formed ((a' ---> b') ---> a ---> b')
wf_auto2.
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
(a' ---> b') ---> a ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a' ---> b',
--------------------------------------
a ---> b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H0" ∶ a' ---> b',
"H1" ∶ a,
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ a ---> a',
"H0" ∶ a' ---> b',
"H1" ∶ a,
--------------------------------------
b'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ a ---> a',
"H0" ∶ a' ---> b',
"H1" ∶ a,
--------------------------------------
a'
using i
Σ: Signature Γ: Theory a, b, a', b': Pattern i: ProofInfo wfa: well_formed a = true wfb: well_formed b = true wfa': well_formed a' = true wfb': well_formed b' = true Haa': Γ ⊢i a <---> a' using i Hbb': Γ ⊢i b <---> b' using i Haa'1: Γ ⊢i a ---> a' using i Haa'2: Γ ⊢i a' ---> a using i Hbb'1: Γ ⊢i b ---> b' using i Hbb'2: Γ ⊢i b' ---> b using i
Γ Ⱶ
"H2" ∶ a ---> a',
"H0" ∶ a' ---> b',
"H1" ∶ a,
--------------------------------------
a
using i
mlExact "H1".Defined.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a and b ---> c using i
→ Γ ⊢i a ---> b ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a and b ---> c using i
→ Γ ⊢i a ---> b ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ ⊢i a ---> b ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed (a ---> b ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ Ⱶ
a ---> b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed (a ---> b ---> c)
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ Ⱶ
a ---> b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ Ⱶ
"H0" ∶ a,
--------------------------------------
b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ Ⱶ
"H0" ∶ a,
"H1" ∶ b,
--------------------------------------
a and b
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ ⊢i a ---> b ---> a and b using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
Γ ⊢i a ---> b ---> a and b using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed a
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed b
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed a
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed b
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a and b ---> c using i
well_formed b
wf_auto2.}Defined.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b ---> c using i
→ Γ ⊢i a and b ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo
well_formed a
→ well_formed b
→ well_formed c
→ Γ ⊢i a ---> b ---> c using i
→ Γ ⊢i a and b ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ ⊢i a and b ---> c using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
well_formed (a and b ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ Ⱶ
a and b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
well_formed (a and b ---> c)
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ Ⱶ
a and b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
well_formed b
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
b
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ Ⱶ
"H0" ∶ a and b,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i
well_formed b
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
b
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ Ⱶ
"H0" ∶ a and b,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
b
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ ⊢i a and b ---> b using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ ⊢i a and b ---> b using BasicReasoning
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
well_formed a
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
well_formed b
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
well_formed b
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b
Γ Ⱶ
"H0" ∶ a and b,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b l1: list named_hypothesis Heql1: l1 = ("H0" ∶ a and b,
) l2: list named_hypothesis Heql2: l2 = ("H1" ∶ b,
)
well_formed a
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
a
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b l1: list named_hypothesis Heql1: l1 = ("H0" ∶ a and b,
) l2: list named_hypothesis Heql2: l2 = ("H1" ∶ b,
)
well_formed a
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
a
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a and b,
--------------------------------------
a
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ ⊢i a and b ---> a using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ ⊢i a and b ---> a using BasicReasoning
apply pf_conj_elim_l; wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
well_formed (b ---> c)
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
"H4" ∶ b ---> c,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a
well_formed (b ---> c)
wf_auto2.
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
"H4" ∶ b ---> c,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
b ---> c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
--------------------------------------
a
using i
mlExact "H2".
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
"H4" ∶ b ---> c,
--------------------------------------
c
using i
Σ: Signature Γ: Theory a, b, c: Pattern i: ProofInfo wfa: well_formed a wfb: well_formed b wfc: well_formed c H: Γ ⊢i a ---> b ---> c using i Hwf: well_formed b Hwf0: well_formed a Hwf1: well_formed (b ---> c)
Γ Ⱶ
"H3" ∶ a ---> b ---> c,
"H0" ∶ a and b,
"H2" ∶ a,
"H1" ∶ b,
"H4" ∶ b ---> c,
--------------------------------------
b
using i
mlExact "H1".Defined.
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern
well_formed a
→ well_formed b
→ Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern
well_formed a
→ well_formed b
→ Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf l
Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf []
Γ
⊢i foldr patt_imp a [] --->
foldr patt_imp b [] --->
foldr patt_imp (a and b) [] using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning
Γ
⊢i foldr patt_imp a (a0 :: l) --->
foldr patt_imp b (a0 :: l) --->
foldr patt_imp (a and b) (a0 :: l)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf []
Γ
⊢i foldr patt_imp a [] --->
foldr patt_imp b [] --->
foldr patt_imp (a and b) [] using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf []
Γ ⊢i a ---> b ---> a and b using BasicReasoning
apply conj_intro; assumption.
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning
Γ
⊢i foldr patt_imp a (a0 :: l) --->
foldr patt_imp b (a0 :: l) --->
foldr patt_imp (a and b) (a0 :: l)
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning
Γ
⊢i (a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning wfl': Pattern.wf (a0 :: l)
Γ
⊢i (a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning wfl': foldr andb true (map well_formed (a0 :: l))
Γ
⊢i (a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning wfl': well_formed a0 &&
foldr andb true (map well_formed l)
Γ
⊢i (a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Pattern.wf l
→ Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ
⊢i (a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ
⊢i (a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
well_formed
((a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l)
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
(a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
well_formed
((a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l)
wf_auto2.
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
(a0 ---> foldr patt_imp a l) --->
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
--------------------------------------
(a0 ---> foldr patt_imp b l) --->
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
--------------------------------------
a0 ---> foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
well_formed (foldr patt_imp a l)
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
--------------------------------------
foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true
well_formed (foldr patt_imp a l)
wf_auto2.
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
--------------------------------------
foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
--------------------------------------
foldr patt_imp a l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
--------------------------------------
a0
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
well_formed (foldr patt_imp b l)
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
foldr patt_imp b l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
"H4" ∶ foldr patt_imp b l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l)
well_formed (foldr patt_imp b l)
wf_auto2.
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
foldr patt_imp b l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l --->
foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
"H4" ∶ foldr patt_imp b l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
foldr patt_imp b l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
--------------------------------------
a0
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H2" ∶ a0,
"H3" ∶ foldr patt_imp a l,
"H4" ∶ foldr patt_imp b l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H1" ∶ a0 ---> foldr patt_imp b l,
"H3" ∶ foldr patt_imp a l,
"H4" ∶ foldr patt_imp b l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H0" ∶ a0 ---> foldr patt_imp a l,
"H3" ∶ foldr patt_imp a l,
"H4" ∶ foldr patt_imp b l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ Ⱶ
"H3" ∶ foldr patt_imp a l,
"H4" ∶ foldr patt_imp b l,
--------------------------------------
foldr patt_imp (a and b) l
using BasicReasoning
Σ: Signature Γ: Theory a, b, a0: Pattern l: list Pattern wfa: well_formed a wfb: well_formed b wfl: Pattern.wf (a0 :: l) IHl: Γ
⊢i foldr patt_imp a l ---> foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning wfa0: well_formed a0 = true wfl': foldr andb true (map well_formed l) = true Hwf: well_formed (foldr patt_imp a l) Hwf0: well_formed (foldr patt_imp b l)
Γ
⊢i foldr patt_imp a l --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning
apply IHl.Defined.
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp a l using i
→ Γ
⊢i foldr patt_imp b l --->
foldr patt_imp (a and b) l using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp a l using i
→ Γ
⊢i foldr patt_imp b l --->
foldr patt_imp (a and b) l using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i
Γ
⊢i foldr patt_imp b l ---> foldr patt_imp (a and b) l
using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l
using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l
using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp b l ---> foldr patt_imp (a and b) l
using BasicReasoning
apply prf_conj_split; assumption.
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i
Γ ⊢i foldr patt_imp a l using i
exact H2.Defined.
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp a l using i
→ Γ ⊢i foldr patt_imp b l using i
→ Γ ⊢i foldr patt_imp (a and b) l using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo
well_formed a
→ well_formed b
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp a l using i
→ Γ ⊢i foldr patt_imp b l using i
→ Γ ⊢i foldr patt_imp (a and b) l using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i H3: Γ ⊢i foldr patt_imp b l using i
Γ ⊢i foldr patt_imp (a and b) l using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i H3: Γ ⊢i foldr patt_imp b l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i H3: Γ ⊢i foldr patt_imp b l using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp (a and b) l using i
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i H3: Γ ⊢i foldr patt_imp b l using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp (a and b) l using i
apply prf_conj_split_meta; assumption.
Σ: Signature Γ: Theory a, b: Pattern l: list Pattern i: ProofInfo H: well_formed a H0: well_formed b H1: Pattern.wf l H2: Γ ⊢i foldr patt_imp a l using i H3: Γ ⊢i foldr patt_imp b l using i
Γ ⊢i foldr patt_imp b l using i
exact H3.Defined.
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo
Γ Ⱶ
l--------------------------------------
a
using i
→ Γ Ⱶ
l--------------------------------------
b
using i
→ Γ Ⱶ
l--------------------------------------
a and b
using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo
Γ Ⱶ
l--------------------------------------
a
using i
→ Γ Ⱶ
l--------------------------------------
b
using i
→ Γ Ⱶ
l--------------------------------------
a and b
using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ Ⱶ
l--------------------------------------
a
using i
Hb: Γ Ⱶ
l--------------------------------------
b
using i
Γ Ⱶ
l--------------------------------------
a and b
using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
a
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
a
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
a
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
a
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
a
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
a
using i
) Hb: well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
b
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
b
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
b
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
b
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
b
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
b
using i
)
well_formed
(mlConclusion
(Γ Ⱶ
l--------------------------------------
a and b
using i
))
→ Pattern.wf
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
a and b
using i
)))
→ mlTheory
(Γ Ⱶ
l--------------------------------------
a and b
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
a and b
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
a and b
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
a and b
using i
)
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i
well_formed (a and b)
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp (a and b) (patterns_of l)
using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed a
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed a
abstract(wf_auto2).
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: well_formed a
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
exact wfl.
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed b
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed b
abstract(wf_auto2).
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: well_formed b
→ Pattern.wf (patterns_of l)
→ Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Pattern.wf (patterns_of l)
exact wfl.
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp (a and b) (patterns_of l) using i
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed a
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed b
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed a
abstract (wf_auto2).
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
well_formed b
Σ: Signature Γ: Theory a, b: Pattern l: hypotheses i: ProofInfo Ha: Γ ⊢i foldr patt_imp a (patterns_of l) using i Hb: Γ ⊢i foldr patt_imp b (patterns_of l) using i wfab: well_formed (a and b) wfl: Pattern.wf (patterns_of l)
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
(a ---> foldr patt_imp (g₁ <---> g₂) l) --->
(a ---> foldr patt_imp g₁ l) <--->
(a ---> foldr patt_imp g₂ l)
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
(a ---> foldr patt_imp g₁ l) <--->
(a ---> foldr patt_imp g₂ l)
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
(a ---> foldr patt_imp g₁ l) --->
a ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
(a ---> foldr patt_imp g₂ l) --->
a ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
(a ---> foldr patt_imp g₁ l) --->
a ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
a ---> foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
(a ---> foldr patt_imp (g₁ <---> g₂) l) --->
a ---> foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
a --->
foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a,
--------------------------------------
foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ foldr patt_imp g₁ l <---> foldr patt_imp g₂ l,
--------------------------------------
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ (foldr patt_imp g₁ l ---> foldr patt_imp g₂ l)
and (foldr patt_imp g₂ l --->
foldr patt_imp g₁ l),
--------------------------------------
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H1" ∶ foldr patt_imp g₁ l ---> foldr patt_imp g₂ l,
"H2" ∶ foldr patt_imp g₂ l ---> foldr patt_imp g₁ l,
--------------------------------------
foldr patt_imp g₁ l ---> foldr patt_imp g₂ l
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
(a ---> foldr patt_imp g₂ l) --->
a ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a ---> foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
a ---> foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
(a ---> foldr patt_imp (g₁ <---> g₂) l) --->
a ---> foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
a --->
foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ a,
--------------------------------------
foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ foldr patt_imp (g₁ <---> g₂) l,
--------------------------------------
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ foldr patt_imp g₁ l <---> foldr patt_imp g₂ l,
--------------------------------------
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H0" ∶ (foldr patt_imp g₁ l ---> foldr patt_imp g₂ l)
and (foldr patt_imp g₂ l --->
foldr patt_imp g₁ l),
--------------------------------------
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂, a: Pattern l: list Pattern wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf (a :: l) IHl: Γ
⊢i foldr patt_imp (g₁ <---> g₂) l --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning wfa: well_formed a = true wfl': foldr andb true (map well_formed l) = true
Γ Ⱶ
"H1" ∶ foldr patt_imp g₁ l ---> foldr patt_imp g₂ l,
"H2" ∶ foldr patt_imp g₂ l ---> foldr patt_imp g₁ l,
--------------------------------------
foldr patt_imp g₂ l ---> foldr patt_imp g₁ l
using BasicReasoning
mlExact "H2".Defined.
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo
well_formed g₁
→ well_formed g₂
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
→ Γ
⊢i foldr patt_imp g₁ l <--->
foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo
well_formed g₁
→ well_formed g₂
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
→ Γ
⊢i foldr patt_imp g₁ l <--->
foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
Γ ⊢i foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
Γ
⊢i ?ϕ₁ --->
foldr patt_imp g₁ l <---> foldr patt_imp g₂ l
using BasicReasoning
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
exact H.Defined.
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo
well_formed g₁
→ well_formed g₂
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
→ Γ ⊢i foldr patt_imp g₁ l using i
→ Γ ⊢i foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo
well_formed g₁
→ well_formed g₂
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
→ Γ ⊢i foldr patt_imp g₁ l using i
→ Γ ⊢i foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed ?ϕ₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i ?ϕ₁ <---> foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i foldr patt_imp ?g₁ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp ?g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed ?g₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed g₂
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Pattern.wf l
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i foldr patt_imp (?g₁ <---> g₂) l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Γ ⊢i foldr patt_imp g₁ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed g₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed g₂
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Pattern.wf l
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed (foldr patt_imp g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed g₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
well_formed g₂
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₁ l using i
Pattern.wf l
all: wf_auto2.Defined.
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo
well_formed g₁
→ well_formed g₂
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
→ Γ ⊢i foldr patt_imp g₂ l using i
→ Γ ⊢i foldr patt_imp g₁ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo
well_formed g₁
→ well_formed g₂
→ Pattern.wf l
→ Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i
→ Γ ⊢i foldr patt_imp g₂ l using i
→ Γ ⊢i foldr patt_imp g₁ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i foldr patt_imp g₁ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g₁ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed ?ϕ₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i foldr patt_imp g₁ l <---> ?ϕ₁using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i foldr patt_imp ?g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp ?g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed g₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed ?g₂
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Pattern.wf l
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i foldr patt_imp (g₁ <---> ?g₂) l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Γ ⊢i foldr patt_imp g₂ l using i
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed g₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed g₂
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
Pattern.wf l
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp g₁ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed (foldr patt_imp g₂ l)
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed g₁
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed g₂
Σ: Signature Γ: Theory g₁, g₂: Pattern l: list Pattern i: ProofInfo wfg₁: well_formed g₁ wfg₂: well_formed g₂ wfl: Pattern.wf l H1: Γ ⊢i foldr patt_imp (g₁ <---> g₂) l using i H2: Γ ⊢i foldr patt_imp g₂ l using i
well_formed x
→ Pattern.wf xs → well_formed (foldr patt_and x xs)
Σ: Signature x: Pattern xs: list Pattern
well_formed x
→ Pattern.wf xs → well_formed (foldr patt_and x xs)
Σ: Signature x: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf xs
well_formed (foldr patt_and x xs)
Σ: Signature x: Pattern wfx: well_formed x wfxs: Pattern.wf []
well_formed x
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
well_formed (a and foldr patt_and x xs)
Σ: Signature x: Pattern wfx: well_formed x wfxs: Pattern.wf []
well_formed x
assumption.
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
well_formed (a and foldr patt_and x xs)
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
Pattern.wf xs
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: well_formed (foldr patt_and x xs)
well_formed (a and foldr patt_and x xs)
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
Pattern.wf xs
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: foldr andb true (map well_formed (a :: xs)) IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
Pattern.wf xs
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: well_formed a &&
foldr andb true (map well_formed xs) IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
Pattern.wf xs
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x H: well_formed a = true H0: foldr andb true (map well_formed xs) = true IHxs: Pattern.wf xs
→ well_formed (foldr patt_and x xs)
Pattern.wf xs
assumption.
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: well_formed (foldr patt_and x xs)
well_formed (a and foldr patt_and x xs)
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: well_formed (foldr patt_and x xs)
well_formed a = true
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: well_formed (foldr patt_and x xs)
well_formed (foldr patt_and x xs) = true
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: well_formed (foldr patt_and x xs)
well_formed a = true
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: foldr andb true (map well_formed (a :: xs)) IHxs: well_formed (foldr patt_and x xs)
well_formed a = true
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: well_formed a &&
foldr andb true (map well_formed xs) IHxs: well_formed (foldr patt_and x xs)
well_formed a = true
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x H: well_formed a = true H0: foldr andb true (map well_formed xs) = true IHxs: well_formed (foldr patt_and x xs)
well_formed a = true
assumption.
Σ: Signature x, a: Pattern xs: list Pattern wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: well_formed (foldr patt_and x xs)
well_formed (foldr patt_and x xs) = true
assumption.Qed.
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern
well_formed g
→ well_formed x
→ Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern
well_formed g
→ well_formed x
→ Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs
Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs) using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf []
Γ ⊢i (x ---> g) ---> x ---> g using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf []
Γ ⊢i (x ---> g) ---> x ---> g using BasicReasoning
Σ: Signature Γ: Theory g, x: Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf []
well_formed (x ---> g)
Σ: Signature Γ: Theory g, x: Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf []
well_formed (x ---> g)
wf_auto2.}
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf (a :: xs) IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfxs: foldr andb true (map well_formed (a :: xs)) IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfxs: well_formed a &&
foldr andb true (map well_formed xs) IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: foldr andb true (map well_formed xs) = true IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Pattern.wf xs
→ Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
foldr patt_imp g (x :: xs)
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
well_formed (foldr patt_and x xs)
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs)
well_formed (foldr patt_and x xs)
apply well_formed_foldr_and; assumption.
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ
⊢i (a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
well_formed
((a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs)
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ Ⱶ
(a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
well_formed
((a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs)
wf_auto2.
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ Ⱶ
(a and foldr patt_and x xs ---> g) --->
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
--------------------------------------
x ---> a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
--------------------------------------
a ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
--------------------------------------
(foldr patt_and x xs ---> g) ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H4" ∶ (foldr patt_and x xs ---> g) --->
foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs)
well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
wf_auto2.
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
--------------------------------------
(foldr patt_and x xs ---> g) ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H4" ∶ (foldr patt_and x xs ---> g) --->
foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
--------------------------------------
(foldr patt_and x xs ---> g) ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
well_formed (x ---> foldr patt_imp g xs)
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
--------------------------------------
x ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
"H6" ∶ x ---> foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
well_formed (x ---> foldr patt_imp g xs)
wf_auto2.
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
--------------------------------------
x ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
"H6" ∶ x ---> foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
--------------------------------------
x ---> foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
--------------------------------------
foldr patt_and x xs ---> g
using BasicReasoning
mlExact "H5".
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
"H6" ∶ x ---> foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
"H6" ∶ x ---> foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs) Hwf0: well_formed (x ---> foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H1" ∶ x,
"H2" ∶ a,
"H5" ∶ foldr patt_and x xs ---> g,
"H6" ∶ x ---> foldr patt_imp g xs,
--------------------------------------
x
using BasicReasoning
mlExact "H1".
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H3" ∶ (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs,
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H4" ∶ (foldr patt_and x xs ---> g) --->
foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H4" ∶ (foldr patt_and x xs ---> g) --->
foldr patt_imp g xs,
--------------------------------------
foldr patt_imp g xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H4" ∶ (foldr patt_and x xs ---> g) --->
foldr patt_imp g xs,
--------------------------------------
foldr patt_and x xs ---> g
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
--------------------------------------
foldr patt_and x xs ---> g
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H3" ∶ foldr patt_and x xs,
--------------------------------------
g
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H3" ∶ foldr patt_and x xs,
--------------------------------------
a and foldr patt_and x xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H3" ∶ foldr patt_and x xs,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H3" ∶ foldr patt_and x xs,
--------------------------------------
foldr patt_and x xs
using BasicReasoning
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H3" ∶ foldr patt_and x xs,
--------------------------------------
a
using BasicReasoning
mlExact "H2".
Σ: Signature Γ: Theory g, x, a: Pattern xs: list Pattern wfg: well_formed g wfx: well_formed x wfa: well_formed a = true wfxs: Pattern.wf xs = true IHxs: Γ
⊢i (foldr patt_and x xs ---> g) --->
x ---> foldr patt_imp g xs
using BasicReasoning wfaxs: Pattern.wf (a :: xs) Hwffa: well_formed (foldr patt_and x xs) Hwf: well_formed
((foldr patt_and x xs ---> g) --->
foldr patt_imp g xs)
Γ Ⱶ
"H0" ∶ a and foldr patt_and x xs ---> g,
"H1" ∶ x,
"H2" ∶ a,
"H3" ∶ foldr patt_and x xs,
--------------------------------------
foldr patt_and x xs
using BasicReasoning
mlExact "H3".Defined.
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo
well_formed g
→ well_formed x
→ Pattern.wf xs
→ Γ ⊢i foldr patt_and x xs ---> g using i
→ Γ ⊢i foldr patt_imp g (x :: xs) using i
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo
well_formed g
→ well_formed x
→ Pattern.wf xs
→ Γ ⊢i foldr patt_and x xs ---> g using i
→ Γ ⊢i foldr patt_imp g (x :: xs) using i
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs H: Γ ⊢i foldr patt_and x xs ---> g using i
Γ ⊢i foldr patt_imp g (x :: xs) using i
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs H: Γ ⊢i foldr patt_and x xs ---> g using i
Γ ⊢i ?ϕ₁using i
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs H: Γ ⊢i foldr patt_and x xs ---> g using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (x :: xs) using i
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs H: Γ ⊢i foldr patt_and x xs ---> g using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (x :: xs) using i
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs H: Γ ⊢i foldr patt_and x xs ---> g using i
Γ ⊢i ?ϕ₁ ---> foldr patt_imp g (x :: xs)
using BasicReasoning
apply lhs_and_to_imp; assumption.
Σ: Signature Γ: Theory g, x: Pattern xs: list Pattern i: ProofInfo wfg: well_formed g wfx: well_formed x wfxs: Pattern.wf xs H: Γ ⊢i foldr patt_and x xs ---> g using i
Γ ⊢i foldr patt_and x xs ---> g using i
exact H.Defined.#[local]
LtactryExact l idx :=
match l with
| nil => idtac
| (?a :: ?m) => try mlExactn idx; tryExact m (idx + 1)
end.#[global]
LtacmlAssumption :=
match goal with
| [ |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm?Ctx?l?g?i) ]
=>
tryExact l 0end.
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i (a ---> b) <---> ! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i (a ---> b) <---> ! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i (a ---> b) <---> ! a or b using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed ((a ---> b) <---> ! a or b)
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
(a ---> b) <---> ! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed ((a ---> b) <---> ! a or b)
wf_auto2.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
(a ---> b) <---> ! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! ((a ---> b) ---> ! a or b)
or ! (! a or b ---> a ---> b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! ((a ---> b) ---> ! a or b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (! a or b ---> a ---> b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! ((a ---> b) ---> ! a or b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ((a ---> b) ---> ! a or b) ---> ⊥,
--------------------------------------
(a ---> b) ---> ! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ((a ---> b) ---> ! a or b) ---> ⊥,
"H2" ∶ a ---> b,
--------------------------------------
! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ a ---> b,
--------------------------------------
! a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ a ---> b,
"H1" ∶ ! ! a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ a ---> b,
"H1" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ a ---> b,
"H1" ∶ a,
--------------------------------------
a
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (! a or b ---> a ---> b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (! a or b ---> a ---> b) ---> ⊥,
--------------------------------------
! a or b ---> a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (! a or b ---> a ---> b) ---> ⊥,
"H0" ∶ ! a or b,
--------------------------------------
a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a or b,
--------------------------------------
a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a or b,
"H1" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H1" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H3" ∶ b,
"H1" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H1" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H1" ∶ a,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ a ---> ⊥,
"H1" ∶ a,
--------------------------------------
a
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H3" ∶ b,
"H1" ∶ a,
--------------------------------------
b
using BasicReasoning
mlAssumption.Defined.
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i ! (a ---> b) <---> a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i ! (a ---> b) <---> a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i ! (a ---> b) <---> a and ! b using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed (! (a ---> b) <---> a and ! b)
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a ---> b) <---> a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed (! (a ---> b) <---> a and ! b)
wf_auto2.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a ---> b) <---> a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! (! (a ---> b) ---> a and ! b)
or ! (a and ! b ---> ! (a ---> b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! (! (a ---> b) ---> a and ! b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (a and ! b ---> ! (a ---> b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! (! (a ---> b) ---> a and ! b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> a and ! b) ---> ⊥,
--------------------------------------
! (a ---> b) ---> a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> a and ! b) ---> ⊥,
"H0" ∶ ! (a ---> b),
--------------------------------------
a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ ! (a ---> b),
--------------------------------------
! (! a or ! ! b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ ! (a ---> b),
"H2" ∶ ! a or ! ! b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H2" ∶ ! a or ! ! b,
--------------------------------------
a ---> b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H2" ∶ ! a or ! ! b,
"H3" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H4" ∶ ! a,
"H3" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H5" ∶ ! ! b,
"H3" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H4" ∶ ! a,
"H3" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H4" ∶ ! a,
"H3" ∶ a,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H4" ∶ a ---> ⊥,
"H3" ∶ a,
--------------------------------------
a
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H5" ∶ ! ! b,
"H3" ∶ a,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ (! (a ---> b) ---> ! (! a or ! ! b)) ---> ⊥,
"H0" ∶ (a ---> b) ---> ⊥,
"H5" ∶ b,
"H3" ∶ a,
--------------------------------------
b
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (a and ! b ---> ! (a ---> b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (a and ! b ---> ! (a ---> b)) ---> ⊥,
--------------------------------------
a and ! b ---> ! (a ---> b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (a and ! b ---> ! (a ---> b)) ---> ⊥,
"H0" ∶ a and ! b,
--------------------------------------
! (a ---> b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (a and ! b ---> ! (a ---> b)) ---> ⊥,
"H0" ∶ a and ! b,
"H1" ∶ a ---> b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (a and ! b ---> ! (a ---> b)) ---> ⊥,
"H3" ∶ a,
"H4" ∶ ! b,
"H1" ∶ a ---> b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (a and ! b ---> ! (a ---> b)) ---> ⊥,
"H3" ∶ a,
"H4" ∶ b ---> ⊥,
"H1" ∶ a ---> b,
--------------------------------------
b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ (a and ! b ---> ! (a ---> b)) ---> ⊥,
"H3" ∶ a,
"H4" ∶ b ---> ⊥,
"H1" ∶ a ---> b,
--------------------------------------
a
using BasicReasoning
mlAssumption.Defined.
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i ! (a and b) <---> ! a or ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i ! (a and b) <---> ! a or ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i ! (a and b) <---> ! a or ! b using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed (! (a and b) <---> ! a or ! b)
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a and b) <---> ! a or ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed (! (a and b) <---> ! a or ! b)
wf_auto2.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a and b) <---> ! a or ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! (! (a and b) ---> ! a or ! b)
or ! (! a or ! b ---> ! (a and b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! (! (a and b) ---> ! a or ! b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (! a or ! b ---> ! (a and b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! (! (a and b) ---> ! a or ! b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (! (a and b) ---> ! a or ! b) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a and b) ---> ! a or ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! (a and b),
--------------------------------------
! a or ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a or ! b,
--------------------------------------
! a or ! b
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (! a or ! b ---> ! (a and b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (! a or ! b ---> ! (a and b)) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! a or ! b ---> ! (a and b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a or ! b,
--------------------------------------
! (a and b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a or ! b,
--------------------------------------
! a or ! b
using BasicReasoning
mlAssumption.Defined.
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i ! (a or b) <---> ! a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern
well_formed a
→ well_formed b
→ Γ ⊢i ! (a or b) <---> ! a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ ⊢i ! (a or b) <---> ! a and ! b using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed (! (a or b) <---> ! a and ! b)
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a or b) <---> ! a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
well_formed (! (a or b) <---> ! a and ! b)
wf_auto2.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a or b) <---> ! a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! (! (a or b) ---> ! a and ! b)
or ! (! a and ! b ---> ! (a or b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! (! (a or b) ---> ! a and ! b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (! a and ! b ---> ! (a or b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H1" ∶ ! (! (a or b) ---> ! a and ! b),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (! (a or b) ---> ! a and ! b) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (a or b) ---> ! a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! (a or b),
--------------------------------------
! a and ! b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! (a or b),
"H1" ∶ ! ! a or ! ! b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H1" ∶ ! ! a or ! ! b,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H2" ∶ ! ! a,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H3" ∶ ! ! b,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H2" ∶ ! ! a,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H2" ∶ a,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H2" ∶ a,
--------------------------------------
a
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H3" ∶ ! ! b,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H3" ∶ b,
--------------------------------------
a or b
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ a or b ---> ⊥,
"H3" ∶ b,
--------------------------------------
b
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! (! a and ! b ---> ! (a or b)),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! (! a and ! b ---> ! (a or b)) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
! a and ! b ---> ! (a or b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a and ! b,
--------------------------------------
! (a or b)
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H0" ∶ ! a and ! b,
"H1" ∶ a or b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H3" ∶ ! b,
"H1" ∶ a or b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H3" ∶ ! b,
"H4" ∶ a,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H3" ∶ ! b,
"H5" ∶ b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H3" ∶ ! b,
"H4" ∶ a,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ a ---> ⊥,
"H3" ∶ ! b,
"H4" ∶ a,
--------------------------------------
a
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H3" ∶ ! b,
"H5" ∶ b,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a, b: Pattern wfa: well_formed a wfb: well_formed b
Γ Ⱶ
"H2" ∶ ! a,
"H3" ∶ b ---> ⊥,
"H5" ∶ b,
--------------------------------------
b
using BasicReasoning
mlAssumption.Defined.
Σ: Signature Γ: Theory a: Pattern
well_formed a
→ Γ ⊢i ! ! a <---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern
well_formed a
→ Γ ⊢i ! ! a <---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ ⊢i ! ! a <---> a using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
well_formed (! ! a <---> a)
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
! ! a <---> a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
well_formed (! ! a <---> a)
wf_auto2.
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
! ! a <---> a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H0" ∶ ! (! ! a ---> a) or ! (a ---> ! ! a),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H1" ∶ ! (! ! a ---> a),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ ! (a ---> ! ! a),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H1" ∶ ! (! ! a ---> a),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H1" ∶ (! ! a ---> a) ---> ⊥,
--------------------------------------
! ! a ---> a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H1" ∶ (! ! a ---> a) ---> ⊥,
"H0" ∶ ! ! a,
--------------------------------------
a
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H1" ∶ (! ! a ---> a) ---> ⊥,
"H0" ∶ a,
--------------------------------------
a
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ ! (a ---> ! ! a),
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ (a ---> (a ---> ⊥) ---> ⊥) ---> ⊥,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ (a ---> (a ---> ⊥) ---> ⊥) ---> ⊥,
--------------------------------------
a ---> (a ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ (a ---> (a ---> ⊥) ---> ⊥) ---> ⊥,
"H0" ∶ a,
--------------------------------------
(a ---> ⊥) ---> ⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ (a ---> (a ---> ⊥) ---> ⊥) ---> ⊥,
"H0" ∶ a,
"H1" ∶ a ---> ⊥,
--------------------------------------
⊥
using BasicReasoning
Σ: Signature Γ: Theory a: Pattern wfa: well_formed a
Γ Ⱶ
"H2" ∶ (a ---> (a ---> ⊥) ---> ⊥) ---> ⊥,
"H0" ∶ a,
"H1" ∶ a ---> ⊥,
--------------------------------------
a
using BasicReasoning
mlAssumption.Defined.
Σ: Signature
∀ (Γ : Theory) (p : Pattern),
well_formed p
→ Γ ⊢i p and p <---> p using BasicReasoning
Σ: Signature
∀ (Γ : Theory) (p : Pattern),
well_formed p
→ Γ ⊢i p and p <---> p using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ ⊢i p and p <---> p using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
well_formed (p and p <---> p)
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
p and p <---> p
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
p and p <---> p
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
"H" ∶ p and p,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
"H" ∶ p,
--------------------------------------
p and p
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
"H" ∶ p and p,
--------------------------------------
p
using BasicReasoning
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
"0" ∶ p,
"1" ∶ p,
--------------------------------------
p
using BasicReasoning
mlAssumption.
Σ: Signature Γ: Theory p: Pattern WFp: well_formed p
Γ Ⱶ
"H" ∶ p,
--------------------------------------
p and p
using BasicReasoning
mlSplitAnd; mlAssumption.Defined.
Σ: Signature
∀ (Γ : Theory) (l : hypotheses) (g : Pattern) (i : ProofInfo),
Γ ⊢i g using i
→ Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature
∀ (Γ : Theory) (l : hypotheses) (g : Pattern) (i : ProofInfo),
Γ ⊢i g using i
→ Γ Ⱶ
l--------------------------------------
g
using i
Σ: Signature Γ: Theory l: hypotheses g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed
(mlConclusion (Γ Ⱶ
l--------------------------------------
g
using i
)) wfl: Pattern.wf
(patterns_of
(mlHypotheses (Γ Ⱶ
l--------------------------------------
g
using i
)))
mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i
)
Σ: Signature Γ: Theory l: hypotheses g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed
(mlConclusion (Γ Ⱶ
l--------------------------------------
g
using i
)) wfl: Pattern.wf
(patterns_of
(mlHypotheses (Γ Ⱶ
l--------------------------------------
g
using i
)))
mlTheory
(Γ Ⱶ
l--------------------------------------
g
using i
)
⊢i foldr patt_imp
(mlConclusion
(Γ Ⱶ
l--------------------------------------
g
using i
))
(patterns_of
(mlHypotheses
(Γ Ⱶ
l--------------------------------------
g
using i
)))
using mlInfo
(Γ Ⱶ
l--------------------------------------
g
using i
)
Σ: Signature Γ: Theory l: hypotheses g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed g wfl: Pattern.wf (patterns_of l)
Γ ⊢i foldr patt_imp g (patterns_of l) using i
Σ: Signature Γ: Theory a: named_hypothesis l: list named_hypothesis g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed g wfl: Pattern.wf (patterns_of (a :: l)) IHl: Pattern.wf (patterns_of l) → Γ ⊢i foldr patt_imp g (patterns_of l) using i
Γ ⊢i nh_patt a ---> foldr patt_imp g (patterns_of l)
using i
Σ: Signature Γ: Theory a: named_hypothesis l: list named_hypothesis g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed g wfl: Pattern.wf (nh_patt a :: patterns_of l) IHl: Pattern.wf (patterns_of l) → Γ ⊢i foldr patt_imp g (patterns_of l) using i
Γ ⊢i nh_patt a ---> foldr patt_imp g (patterns_of l)
using i
Σ: Signature Γ: Theory a: named_hypothesis l: list named_hypothesis g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed g wfl: Pattern.wf (nh_patt a :: patterns_of l) IHl: Pattern.wf (patterns_of l) → Γ ⊢i foldr patt_imp g (patterns_of l) using i wfl': Pattern.wf (patterns_of l) = true
Γ ⊢i nh_patt a ---> foldr patt_imp g (patterns_of l)
using i
Σ: Signature Γ: Theory a: named_hypothesis l: list named_hypothesis g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed g wfl: well_formed (nh_patt a) && foldr andb true (map well_formed (patterns_of l)) IHl: Pattern.wf (patterns_of l) → Γ ⊢i foldr patt_imp g (patterns_of l) using i wfl': Pattern.wf (patterns_of l) = true
Γ ⊢i nh_patt a ---> foldr patt_imp g (patterns_of l)
using i
Σ: Signature Γ: Theory a: named_hypothesis l: list named_hypothesis g: Pattern i: ProofInfo pf: Γ ⊢i g using i wfG: well_formed g wfl: well_formed (nh_patt a) = true IHl: Pattern.wf (patterns_of l) → Γ ⊢i foldr patt_imp g (patterns_of l) using i wfl': Pattern.wf (patterns_of l) = true
Γ ⊢i nh_patt a ---> foldr patt_imp g (patterns_of l)
using i
∀ (Σ : Signature) (Γ : Theory),
Γ ⊢i Top using BasicReasoning
∀ (Σ : Signature) (Γ : Theory),
Γ ⊢i Top using BasicReasoning
Σ: Signature Γ: Theory
Γ ⊢i Top using BasicReasoning
Σ: Signature Γ: Theory
well_formed Top
Σ: Signature Γ: Theory
Γ Ⱶ
Top
using BasicReasoning
Σ: Signature Γ: Theory
Γ Ⱶ
Top
using BasicReasoning
mlExactMeta (top_holds Γ).Defined.(**********************************************************************************)(* This is an example and belongs to the end of this file. Its only purpose is only to show as many tactics as possible.\ *)
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p <---> p' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p <---> p' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q <---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q <---> q' using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p and q <---> p' and q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p and q <---> p' and q')
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p and q <---> p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p and q <---> p' and q')
wf_auto2.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p and q <---> p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
(p and q ---> p' and q') and (p' and q' ---> p and q)
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p and q ---> p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' and q' ---> p and q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p and q ---> p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p and q,
--------------------------------------
p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q,
--------------------------------------
p' and q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q,
--------------------------------------
p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q,
--------------------------------------
q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q,
--------------------------------------
p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q,
--------------------------------------
p
using i
mlExactn 0.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q,
--------------------------------------
q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
"H2" ∶ q',
--------------------------------------
q'
using i
mlExact "H2".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' and q' ---> p and q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
--------------------------------------
p and q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
--------------------------------------
! (! p or ! q)
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
--------------------------------------
! p or ! q ---> ⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
"H1" ∶ ! p or ! q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
"H2" ∶ ! p,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
"H3" ∶ ! q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
"H2" ∶ ! p,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H3" ∶ p',
"H4" ∶ q',
"H2" ∶ ! p,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H3" ∶ p',
"H4" ∶ q',
"H2" ∶ p ---> ⊥,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H3" ∶ p',
"H4" ∶ q',
"H2" ∶ p ---> ⊥,
--------------------------------------
p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H3" ∶ p',
"H4" ∶ q',
--------------------------------------
p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H3" ∶ p',
--------------------------------------
p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p' ---> p using i
exact p'ip.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' and q',
"H3" ∶ ! q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ q' ---> q,
"H0" ∶ p' and q',
"H3" ∶ ! q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed q
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
--------------------------------------
q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
"Hq" ∶ q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed q
wf_auto2.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
--------------------------------------
q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
"Hq" ∶ q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
--------------------------------------
q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
--------------------------------------
q'
using i
mlExact "H4".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ ! q,
"Hq" ∶ q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ q ---> ⊥,
"Hq" ∶ q,
--------------------------------------
⊥
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i Hwf: well_formed q
Γ Ⱶ
"H1" ∶ q' ---> q,
"H2" ∶ p',
"H4" ∶ q',
"H3" ∶ q ---> ⊥,
"Hq" ∶ q,
--------------------------------------
q
using i
mlAssumption.Defined.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo
well_formed p
→ well_formed q
→ well_formed p'
→ well_formed q'
→ Γ ⊢i p <---> p' using i
→ Γ ⊢i q <---> q' using i
→ Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p <---> p' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p <---> p' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q <---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q <---> q' using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ ⊢i p or q <---> p' or q' using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p or q <---> p' or q')
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p or q <---> p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
well_formed (p or q <---> p' or q')
wf_auto2.
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p or q <---> p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
(p or q ---> p' or q') and (p' or q' ---> p or q)
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p or q ---> p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' or q' ---> p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p or q ---> p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p or q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p,
--------------------------------------
p
using i
mlExact "H1".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
p' or q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
q'
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q,
--------------------------------------
q
using i
mlExact "H2".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
p' or q' ---> p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H0" ∶ p' or q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H1" ∶ p',
--------------------------------------
p'
using i
mlExact "H1".
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
p or q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
q
using i
Σ: Signature Γ: Theory p, q, p', q': Pattern i: ProofInfo wfp: well_formed p wfq: well_formed q wfp': well_formed p' wfq': well_formed q' pep': Γ ⊢i p <---> p' using i qeq': Γ ⊢i q <---> q' using i pip': Γ ⊢i p ---> p' using i p'ip: Γ ⊢i p' ---> p using i qiq': Γ ⊢i q ---> q' using i q'iq: Γ ⊢i q' ---> q using i
Γ Ⱶ
"H2" ∶ q',
--------------------------------------
q'
using i